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1 | 2718-2721 | 8 Find the values of x for which y = [x(x – 2)]2 is an increasing function 9 Prove that
(24sin
cos )
y
θ
=
− θ
+
θ
is an increasing function of θ in 0 2
, π
|
1 | 2719-2722 | Find the values of x for which y = [x(x – 2)]2 is an increasing function 9 Prove that
(24sin
cos )
y
θ
=
− θ
+
θ
is an increasing function of θ in 0 2
, π
Rationalised 2023-24
APPLICATION OF DERIVATIVES
159
10 |
1 | 2720-2723 | 9 Prove that
(24sin
cos )
y
θ
=
− θ
+
θ
is an increasing function of θ in 0 2
, π
Rationalised 2023-24
APPLICATION OF DERIVATIVES
159
10 Prove that the logarithmic function is increasing on (0, ∞) |
1 | 2721-2724 | Prove that
(24sin
cos )
y
θ
=
− θ
+
θ
is an increasing function of θ in 0 2
, π
Rationalised 2023-24
APPLICATION OF DERIVATIVES
159
10 Prove that the logarithmic function is increasing on (0, ∞) 11 |
1 | 2722-2725 | Rationalised 2023-24
APPLICATION OF DERIVATIVES
159
10 Prove that the logarithmic function is increasing on (0, ∞) 11 Prove that the function f given by f(x) = x2 – x + 1 is neither strictly increasing
nor decreasing on (– 1, 1) |
1 | 2723-2726 | Prove that the logarithmic function is increasing on (0, ∞) 11 Prove that the function f given by f(x) = x2 – x + 1 is neither strictly increasing
nor decreasing on (– 1, 1) 12 |
1 | 2724-2727 | 11 Prove that the function f given by f(x) = x2 – x + 1 is neither strictly increasing
nor decreasing on (– 1, 1) 12 Which of the following functions are decreasing on 0, 2
π |
1 | 2725-2728 | Prove that the function f given by f(x) = x2 – x + 1 is neither strictly increasing
nor decreasing on (– 1, 1) 12 Which of the following functions are decreasing on 0, 2
π (A) cos x
(B) cos 2x
(C) cos 3x
(D) tan x
13 |
1 | 2726-2729 | 12 Which of the following functions are decreasing on 0, 2
π (A) cos x
(B) cos 2x
(C) cos 3x
(D) tan x
13 On which of the following intervals is the function f given by f (x) = x100 + sin x –1
decreasing |
1 | 2727-2730 | Which of the following functions are decreasing on 0, 2
π (A) cos x
(B) cos 2x
(C) cos 3x
(D) tan x
13 On which of the following intervals is the function f given by f (x) = x100 + sin x –1
decreasing (A) (0,1)
(B)
π2,
π
(C)
0, 2
π
(D) None of these
14 |
1 | 2728-2731 | (A) cos x
(B) cos 2x
(C) cos 3x
(D) tan x
13 On which of the following intervals is the function f given by f (x) = x100 + sin x –1
decreasing (A) (0,1)
(B)
π2,
π
(C)
0, 2
π
(D) None of these
14 For what values of a the function f given by f (x) = x2 + ax + 1 is increasing on
[1, 2] |
1 | 2729-2732 | On which of the following intervals is the function f given by f (x) = x100 + sin x –1
decreasing (A) (0,1)
(B)
π2,
π
(C)
0, 2
π
(D) None of these
14 For what values of a the function f given by f (x) = x2 + ax + 1 is increasing on
[1, 2] 15 |
1 | 2730-2733 | (A) (0,1)
(B)
π2,
π
(C)
0, 2
π
(D) None of these
14 For what values of a the function f given by f (x) = x2 + ax + 1 is increasing on
[1, 2] 15 Let I be any interval disjoint from [–1, 1] |
1 | 2731-2734 | For what values of a the function f given by f (x) = x2 + ax + 1 is increasing on
[1, 2] 15 Let I be any interval disjoint from [–1, 1] Prove that the function f given by
1
( )
f x
x
x
=
+
is increasing on I |
1 | 2732-2735 | 15 Let I be any interval disjoint from [–1, 1] Prove that the function f given by
1
( )
f x
x
x
=
+
is increasing on I 16 |
1 | 2733-2736 | Let I be any interval disjoint from [–1, 1] Prove that the function f given by
1
( )
f x
x
x
=
+
is increasing on I 16 Prove that the function f given by f (x) = log sin x is increasing on 0 2
, π
and
decreasing on π π
2 ,
|
1 | 2734-2737 | Prove that the function f given by
1
( )
f x
x
x
=
+
is increasing on I 16 Prove that the function f given by f (x) = log sin x is increasing on 0 2
, π
and
decreasing on π π
2 ,
17 |
1 | 2735-2738 | 16 Prove that the function f given by f (x) = log sin x is increasing on 0 2
, π
and
decreasing on π π
2 ,
17 Prove that the function f given by f (x) = log |cos x| is decreasing on 0, 2
π
and
increasing on 3 , 2
π2
π
|
1 | 2736-2739 | Prove that the function f given by f (x) = log sin x is increasing on 0 2
, π
and
decreasing on π π
2 ,
17 Prove that the function f given by f (x) = log |cos x| is decreasing on 0, 2
π
and
increasing on 3 , 2
π2
π
18 |
1 | 2737-2740 | 17 Prove that the function f given by f (x) = log |cos x| is decreasing on 0, 2
π
and
increasing on 3 , 2
π2
π
18 Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R |
1 | 2738-2741 | Prove that the function f given by f (x) = log |cos x| is decreasing on 0, 2
π
and
increasing on 3 , 2
π2
π
18 Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R 19 |
1 | 2739-2742 | 18 Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R 19 The interval in which y = x2 e–x is increasing is
(A) (– ∞, ∞)
(B) (– 2, 0)
(C) (2, ∞)
(D) (0, 2)
6 |
1 | 2740-2743 | Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R 19 The interval in which y = x2 e–x is increasing is
(A) (– ∞, ∞)
(B) (– 2, 0)
(C) (2, ∞)
(D) (0, 2)
6 4 Maxima and Minima
In this section, we will use the concept of derivatives to calculate the maximum or
minimum values of various functions |
1 | 2741-2744 | 19 The interval in which y = x2 e–x is increasing is
(A) (– ∞, ∞)
(B) (– 2, 0)
(C) (2, ∞)
(D) (0, 2)
6 4 Maxima and Minima
In this section, we will use the concept of derivatives to calculate the maximum or
minimum values of various functions In fact, we will find the ‘turning points’ of the
graph of a function and thus find points at which the graph reaches its highest (or
Rationalised 2023-24
MATHEMATICS
160
lowest) locally |
1 | 2742-2745 | The interval in which y = x2 e–x is increasing is
(A) (– ∞, ∞)
(B) (– 2, 0)
(C) (2, ∞)
(D) (0, 2)
6 4 Maxima and Minima
In this section, we will use the concept of derivatives to calculate the maximum or
minimum values of various functions In fact, we will find the ‘turning points’ of the
graph of a function and thus find points at which the graph reaches its highest (or
Rationalised 2023-24
MATHEMATICS
160
lowest) locally The knowledge of such points is very useful in sketching the graph of
a given function |
1 | 2743-2746 | 4 Maxima and Minima
In this section, we will use the concept of derivatives to calculate the maximum or
minimum values of various functions In fact, we will find the ‘turning points’ of the
graph of a function and thus find points at which the graph reaches its highest (or
Rationalised 2023-24
MATHEMATICS
160
lowest) locally The knowledge of such points is very useful in sketching the graph of
a given function Further, we will also find the absolute maximum and absolute minimum
of a function that are necessary for the solution of many applied problems |
1 | 2744-2747 | In fact, we will find the ‘turning points’ of the
graph of a function and thus find points at which the graph reaches its highest (or
Rationalised 2023-24
MATHEMATICS
160
lowest) locally The knowledge of such points is very useful in sketching the graph of
a given function Further, we will also find the absolute maximum and absolute minimum
of a function that are necessary for the solution of many applied problems (i)Let us consider the following problems that arise in day to day life |
1 | 2745-2748 | The knowledge of such points is very useful in sketching the graph of
a given function Further, we will also find the absolute maximum and absolute minimum
of a function that are necessary for the solution of many applied problems (i)Let us consider the following problems that arise in day to day life The profit from a grove of orange trees is given by P(x) = ax + bx2, where a,b
are constants and x is the number of orange trees per acre |
1 | 2746-2749 | Further, we will also find the absolute maximum and absolute minimum
of a function that are necessary for the solution of many applied problems (i)Let us consider the following problems that arise in day to day life The profit from a grove of orange trees is given by P(x) = ax + bx2, where a,b
are constants and x is the number of orange trees per acre How many trees per
acre will maximise the profit |
1 | 2747-2750 | (i)Let us consider the following problems that arise in day to day life The profit from a grove of orange trees is given by P(x) = ax + bx2, where a,b
are constants and x is the number of orange trees per acre How many trees per
acre will maximise the profit (ii)
A ball, thrown into the air from a building 60 metres high, travels along a path
given by
2
( )
60
x60
h x
x
=
+
−
, where x is the horizontal distance from the building
and h(x) is the height of the ball |
1 | 2748-2751 | The profit from a grove of orange trees is given by P(x) = ax + bx2, where a,b
are constants and x is the number of orange trees per acre How many trees per
acre will maximise the profit (ii)
A ball, thrown into the air from a building 60 metres high, travels along a path
given by
2
( )
60
x60
h x
x
=
+
−
, where x is the horizontal distance from the building
and h(x) is the height of the ball What is the maximum height the ball will
reach |
1 | 2749-2752 | How many trees per
acre will maximise the profit (ii)
A ball, thrown into the air from a building 60 metres high, travels along a path
given by
2
( )
60
x60
h x
x
=
+
−
, where x is the horizontal distance from the building
and h(x) is the height of the ball What is the maximum height the ball will
reach (iii)
An Apache helicopter of enemy is flying along the path given by the curve
f (x) = x2 + 7 |
1 | 2750-2753 | (ii)
A ball, thrown into the air from a building 60 metres high, travels along a path
given by
2
( )
60
x60
h x
x
=
+
−
, where x is the horizontal distance from the building
and h(x) is the height of the ball What is the maximum height the ball will
reach (iii)
An Apache helicopter of enemy is flying along the path given by the curve
f (x) = x2 + 7 A soldier, placed at the point (1, 2), wants to shoot the helicopter
when it is nearest to him |
1 | 2751-2754 | What is the maximum height the ball will
reach (iii)
An Apache helicopter of enemy is flying along the path given by the curve
f (x) = x2 + 7 A soldier, placed at the point (1, 2), wants to shoot the helicopter
when it is nearest to him What is the nearest distance |
1 | 2752-2755 | (iii)
An Apache helicopter of enemy is flying along the path given by the curve
f (x) = x2 + 7 A soldier, placed at the point (1, 2), wants to shoot the helicopter
when it is nearest to him What is the nearest distance In each of the above problem, there is something common, i |
1 | 2753-2756 | A soldier, placed at the point (1, 2), wants to shoot the helicopter
when it is nearest to him What is the nearest distance In each of the above problem, there is something common, i e |
1 | 2754-2757 | What is the nearest distance In each of the above problem, there is something common, i e , we wish to find out
the maximum or minimum values of the given functions |
1 | 2755-2758 | In each of the above problem, there is something common, i e , we wish to find out
the maximum or minimum values of the given functions In order to tackle such problems,
we first formally define maximum or minimum values of a function, points of local
maxima and minima and test for determining such points |
1 | 2756-2759 | e , we wish to find out
the maximum or minimum values of the given functions In order to tackle such problems,
we first formally define maximum or minimum values of a function, points of local
maxima and minima and test for determining such points Definition 3 Let f be a function defined on an interval I |
1 | 2757-2760 | , we wish to find out
the maximum or minimum values of the given functions In order to tackle such problems,
we first formally define maximum or minimum values of a function, points of local
maxima and minima and test for determining such points Definition 3 Let f be a function defined on an interval I Then
(a)
f is said to have a maximum value in I, if there exists a point c in I such that
( )
( )
f c>
f x , for all x ∈ I |
1 | 2758-2761 | In order to tackle such problems,
we first formally define maximum or minimum values of a function, points of local
maxima and minima and test for determining such points Definition 3 Let f be a function defined on an interval I Then
(a)
f is said to have a maximum value in I, if there exists a point c in I such that
( )
( )
f c>
f x , for all x ∈ I The number f (c) is called the maximum value of f in I and the point c is called a
point of maximum value of f in I |
1 | 2759-2762 | Definition 3 Let f be a function defined on an interval I Then
(a)
f is said to have a maximum value in I, if there exists a point c in I such that
( )
( )
f c>
f x , for all x ∈ I The number f (c) is called the maximum value of f in I and the point c is called a
point of maximum value of f in I (b)
f is said to have a minimum value in I, if there exists a point c in I such that
f (c) < f (x), for all x ∈ I |
1 | 2760-2763 | Then
(a)
f is said to have a maximum value in I, if there exists a point c in I such that
( )
( )
f c>
f x , for all x ∈ I The number f (c) is called the maximum value of f in I and the point c is called a
point of maximum value of f in I (b)
f is said to have a minimum value in I, if there exists a point c in I such that
f (c) < f (x), for all x ∈ I The number f (c), in this case, is called the minimum value of f in I and the point
c, in this case, is called a point of minimum value of f in I |
1 | 2761-2764 | The number f (c) is called the maximum value of f in I and the point c is called a
point of maximum value of f in I (b)
f is said to have a minimum value in I, if there exists a point c in I such that
f (c) < f (x), for all x ∈ I The number f (c), in this case, is called the minimum value of f in I and the point
c, in this case, is called a point of minimum value of f in I (c)
f is said to have an extreme value in I if there exists a point c in I such that
f (c) is either a maximum value or a minimum value of f in I |
1 | 2762-2765 | (b)
f is said to have a minimum value in I, if there exists a point c in I such that
f (c) < f (x), for all x ∈ I The number f (c), in this case, is called the minimum value of f in I and the point
c, in this case, is called a point of minimum value of f in I (c)
f is said to have an extreme value in I if there exists a point c in I such that
f (c) is either a maximum value or a minimum value of f in I The number f (c), in this case, is called an extreme value of f in I and the point c
is called an extreme point |
1 | 2763-2766 | The number f (c), in this case, is called the minimum value of f in I and the point
c, in this case, is called a point of minimum value of f in I (c)
f is said to have an extreme value in I if there exists a point c in I such that
f (c) is either a maximum value or a minimum value of f in I The number f (c), in this case, is called an extreme value of f in I and the point c
is called an extreme point Remark In Fig 6 |
1 | 2764-2767 | (c)
f is said to have an extreme value in I if there exists a point c in I such that
f (c) is either a maximum value or a minimum value of f in I The number f (c), in this case, is called an extreme value of f in I and the point c
is called an extreme point Remark In Fig 6 7(a), (b) and (c), we have exhibited that graphs of certain particular
functions help us to find maximum value and minimum value at a point |
1 | 2765-2768 | The number f (c), in this case, is called an extreme value of f in I and the point c
is called an extreme point Remark In Fig 6 7(a), (b) and (c), we have exhibited that graphs of certain particular
functions help us to find maximum value and minimum value at a point Infact, through
graphs, we can even find maximum/minimum value of a function at a point at which it
is not even differentiable (Example 15) |
1 | 2766-2769 | Remark In Fig 6 7(a), (b) and (c), we have exhibited that graphs of certain particular
functions help us to find maximum value and minimum value at a point Infact, through
graphs, we can even find maximum/minimum value of a function at a point at which it
is not even differentiable (Example 15) Rationalised 2023-24
APPLICATION OF DERIVATIVES
161
Fig 6 |
1 | 2767-2770 | 7(a), (b) and (c), we have exhibited that graphs of certain particular
functions help us to find maximum value and minimum value at a point Infact, through
graphs, we can even find maximum/minimum value of a function at a point at which it
is not even differentiable (Example 15) Rationalised 2023-24
APPLICATION OF DERIVATIVES
161
Fig 6 7
Example 14 Find the maximum and the minimum values, if
any, of the function f given by
f (x) = x2, x ∈ R |
1 | 2768-2771 | Infact, through
graphs, we can even find maximum/minimum value of a function at a point at which it
is not even differentiable (Example 15) Rationalised 2023-24
APPLICATION OF DERIVATIVES
161
Fig 6 7
Example 14 Find the maximum and the minimum values, if
any, of the function f given by
f (x) = x2, x ∈ R Solution From the graph of the given function (Fig 6 |
1 | 2769-2772 | Rationalised 2023-24
APPLICATION OF DERIVATIVES
161
Fig 6 7
Example 14 Find the maximum and the minimum values, if
any, of the function f given by
f (x) = x2, x ∈ R Solution From the graph of the given function (Fig 6 8), we
have f (x) = 0 if x = 0 |
1 | 2770-2773 | 7
Example 14 Find the maximum and the minimum values, if
any, of the function f given by
f (x) = x2, x ∈ R Solution From the graph of the given function (Fig 6 8), we
have f (x) = 0 if x = 0 Also
f (x) ≥ 0, for all x ∈ R |
1 | 2771-2774 | Solution From the graph of the given function (Fig 6 8), we
have f (x) = 0 if x = 0 Also
f (x) ≥ 0, for all x ∈ R Therefore, the minimum value of f is 0 and the point of
minimum value of f is x = 0 |
1 | 2772-2775 | 8), we
have f (x) = 0 if x = 0 Also
f (x) ≥ 0, for all x ∈ R Therefore, the minimum value of f is 0 and the point of
minimum value of f is x = 0 Further, it may be observed
from the graph of the function that f has no maximum value
and hence no point of maximum value of f in R |
1 | 2773-2776 | Also
f (x) ≥ 0, for all x ∈ R Therefore, the minimum value of f is 0 and the point of
minimum value of f is x = 0 Further, it may be observed
from the graph of the function that f has no maximum value
and hence no point of maximum value of f in R ANote If we restrict the domain of f to [– 2, 1] only,
then f will have maximum value(– 2)2 = 4 at x = – 2 |
1 | 2774-2777 | Therefore, the minimum value of f is 0 and the point of
minimum value of f is x = 0 Further, it may be observed
from the graph of the function that f has no maximum value
and hence no point of maximum value of f in R ANote If we restrict the domain of f to [– 2, 1] only,
then f will have maximum value(– 2)2 = 4 at x = – 2 Example 15 Find the maximum and minimum values
of f , if any, of the function given by f(x) = |x|, x ∈ R |
1 | 2775-2778 | Further, it may be observed
from the graph of the function that f has no maximum value
and hence no point of maximum value of f in R ANote If we restrict the domain of f to [– 2, 1] only,
then f will have maximum value(– 2)2 = 4 at x = – 2 Example 15 Find the maximum and minimum values
of f , if any, of the function given by f(x) = |x|, x ∈ R Solution From the graph of the given function
(Fig 6 |
1 | 2776-2779 | ANote If we restrict the domain of f to [– 2, 1] only,
then f will have maximum value(– 2)2 = 4 at x = – 2 Example 15 Find the maximum and minimum values
of f , if any, of the function given by f(x) = |x|, x ∈ R Solution From the graph of the given function
(Fig 6 9) , note that
f (x) ≥ 0, for all x ∈ R and f (x) = 0 if x = 0 |
1 | 2777-2780 | Example 15 Find the maximum and minimum values
of f , if any, of the function given by f(x) = |x|, x ∈ R Solution From the graph of the given function
(Fig 6 9) , note that
f (x) ≥ 0, for all x ∈ R and f (x) = 0 if x = 0 Therefore, the function f has a minimum value 0
and the point of minimum value of f is x = 0 |
1 | 2778-2781 | Solution From the graph of the given function
(Fig 6 9) , note that
f (x) ≥ 0, for all x ∈ R and f (x) = 0 if x = 0 Therefore, the function f has a minimum value 0
and the point of minimum value of f is x = 0 Also, the
graph clearly shows that f has no maximum value in R
and hence no point of maximum value in R |
1 | 2779-2782 | 9) , note that
f (x) ≥ 0, for all x ∈ R and f (x) = 0 if x = 0 Therefore, the function f has a minimum value 0
and the point of minimum value of f is x = 0 Also, the
graph clearly shows that f has no maximum value in R
and hence no point of maximum value in R ANote
(i)
If we restrict the domain of f to [– 2, 1] only, then f will have maximum value
|– 2| = 2 |
1 | 2780-2783 | Therefore, the function f has a minimum value 0
and the point of minimum value of f is x = 0 Also, the
graph clearly shows that f has no maximum value in R
and hence no point of maximum value in R ANote
(i)
If we restrict the domain of f to [– 2, 1] only, then f will have maximum value
|– 2| = 2 Fig 6 |
1 | 2781-2784 | Also, the
graph clearly shows that f has no maximum value in R
and hence no point of maximum value in R ANote
(i)
If we restrict the domain of f to [– 2, 1] only, then f will have maximum value
|– 2| = 2 Fig 6 8
Fig 6 |
1 | 2782-2785 | ANote
(i)
If we restrict the domain of f to [– 2, 1] only, then f will have maximum value
|– 2| = 2 Fig 6 8
Fig 6 9
Rationalised 2023-24
MATHEMATICS
162
Fig 6 |
1 | 2783-2786 | Fig 6 8
Fig 6 9
Rationalised 2023-24
MATHEMATICS
162
Fig 6 10
(ii)
One may note that the function f in Example 27 is not differentiable at
x = 0 |
1 | 2784-2787 | 8
Fig 6 9
Rationalised 2023-24
MATHEMATICS
162
Fig 6 10
(ii)
One may note that the function f in Example 27 is not differentiable at
x = 0 Example 16 Find the maximum and the minimum values, if any, of the function
given by
f (x) = x, x ∈ (0, 1) |
1 | 2785-2788 | 9
Rationalised 2023-24
MATHEMATICS
162
Fig 6 10
(ii)
One may note that the function f in Example 27 is not differentiable at
x = 0 Example 16 Find the maximum and the minimum values, if any, of the function
given by
f (x) = x, x ∈ (0, 1) Solution The given function is an increasing (strictly) function in the given interval
(0, 1) |
1 | 2786-2789 | 10
(ii)
One may note that the function f in Example 27 is not differentiable at
x = 0 Example 16 Find the maximum and the minimum values, if any, of the function
given by
f (x) = x, x ∈ (0, 1) Solution The given function is an increasing (strictly) function in the given interval
(0, 1) From the graph (Fig 6 |
1 | 2787-2790 | Example 16 Find the maximum and the minimum values, if any, of the function
given by
f (x) = x, x ∈ (0, 1) Solution The given function is an increasing (strictly) function in the given interval
(0, 1) From the graph (Fig 6 10) of the function f , it
seems that, it should have the minimum value at a
point closest to 0 on its right and the maximum value
at a point closest to 1 on its left |
1 | 2788-2791 | Solution The given function is an increasing (strictly) function in the given interval
(0, 1) From the graph (Fig 6 10) of the function f , it
seems that, it should have the minimum value at a
point closest to 0 on its right and the maximum value
at a point closest to 1 on its left Are such points
available |
1 | 2789-2792 | From the graph (Fig 6 10) of the function f , it
seems that, it should have the minimum value at a
point closest to 0 on its right and the maximum value
at a point closest to 1 on its left Are such points
available Of course, not |
1 | 2790-2793 | 10) of the function f , it
seems that, it should have the minimum value at a
point closest to 0 on its right and the maximum value
at a point closest to 1 on its left Are such points
available Of course, not It is not possible to locate
such points |
1 | 2791-2794 | Are such points
available Of course, not It is not possible to locate
such points Infact, if a point x0 is closest to 0, then
we find
0
0
x2
<x
for all
0
x ∈(0,1) |
1 | 2792-2795 | Of course, not It is not possible to locate
such points Infact, if a point x0 is closest to 0, then
we find
0
0
x2
<x
for all
0
x ∈(0,1) Also, if x1 is closest
to 1, then
1
1
x21
x
+
>
for all
1
x ∈(0,1) |
1 | 2793-2796 | It is not possible to locate
such points Infact, if a point x0 is closest to 0, then
we find
0
0
x2
<x
for all
0
x ∈(0,1) Also, if x1 is closest
to 1, then
1
1
x21
x
+
>
for all
1
x ∈(0,1) Therefore, the given function has neither the
maximum value nor the minimum value in the interval (0,1) |
1 | 2794-2797 | Infact, if a point x0 is closest to 0, then
we find
0
0
x2
<x
for all
0
x ∈(0,1) Also, if x1 is closest
to 1, then
1
1
x21
x
+
>
for all
1
x ∈(0,1) Therefore, the given function has neither the
maximum value nor the minimum value in the interval (0,1) Remark The reader may observe that in Example 16, if we include the points 0 and 1
in the domain of f , i |
1 | 2795-2798 | Also, if x1 is closest
to 1, then
1
1
x21
x
+
>
for all
1
x ∈(0,1) Therefore, the given function has neither the
maximum value nor the minimum value in the interval (0,1) Remark The reader may observe that in Example 16, if we include the points 0 and 1
in the domain of f , i e |
1 | 2796-2799 | Therefore, the given function has neither the
maximum value nor the minimum value in the interval (0,1) Remark The reader may observe that in Example 16, if we include the points 0 and 1
in the domain of f , i e , if we extend the domain of f to [0,1], then the function f has
minimum value 0 at x = 0 and maximum value 1 at x = 1 |
1 | 2797-2800 | Remark The reader may observe that in Example 16, if we include the points 0 and 1
in the domain of f , i e , if we extend the domain of f to [0,1], then the function f has
minimum value 0 at x = 0 and maximum value 1 at x = 1 Infact, we have the following
results (The proof of these results are beyond the scope of the present text)
Every monotonic function assumes its maximum/minimum value at the end
points of the domain of definition of the function |
1 | 2798-2801 | e , if we extend the domain of f to [0,1], then the function f has
minimum value 0 at x = 0 and maximum value 1 at x = 1 Infact, we have the following
results (The proof of these results are beyond the scope of the present text)
Every monotonic function assumes its maximum/minimum value at the end
points of the domain of definition of the function A more general result is
Every continuous function on a closed interval has a maximum and a minimum
value |
1 | 2799-2802 | , if we extend the domain of f to [0,1], then the function f has
minimum value 0 at x = 0 and maximum value 1 at x = 1 Infact, we have the following
results (The proof of these results are beyond the scope of the present text)
Every monotonic function assumes its maximum/minimum value at the end
points of the domain of definition of the function A more general result is
Every continuous function on a closed interval has a maximum and a minimum
value ANote By a monotonic function f in an interval I, we mean that f is either
increasing in I or decreasing in I |
1 | 2800-2803 | Infact, we have the following
results (The proof of these results are beyond the scope of the present text)
Every monotonic function assumes its maximum/minimum value at the end
points of the domain of definition of the function A more general result is
Every continuous function on a closed interval has a maximum and a minimum
value ANote By a monotonic function f in an interval I, we mean that f is either
increasing in I or decreasing in I Maximum and minimum values of a function defined on a closed interval will be
discussed later in this section |
1 | 2801-2804 | A more general result is
Every continuous function on a closed interval has a maximum and a minimum
value ANote By a monotonic function f in an interval I, we mean that f is either
increasing in I or decreasing in I Maximum and minimum values of a function defined on a closed interval will be
discussed later in this section Let us now examine the graph of a function as shown in Fig 6 |
1 | 2802-2805 | ANote By a monotonic function f in an interval I, we mean that f is either
increasing in I or decreasing in I Maximum and minimum values of a function defined on a closed interval will be
discussed later in this section Let us now examine the graph of a function as shown in Fig 6 11 |
1 | 2803-2806 | Maximum and minimum values of a function defined on a closed interval will be
discussed later in this section Let us now examine the graph of a function as shown in Fig 6 11 Observe that at
points A, B, C and D on the graph, the function changes its nature from decreasing to
increasing or vice-versa |
1 | 2804-2807 | Let us now examine the graph of a function as shown in Fig 6 11 Observe that at
points A, B, C and D on the graph, the function changes its nature from decreasing to
increasing or vice-versa These points may be called turning points of the given
function |
1 | 2805-2808 | 11 Observe that at
points A, B, C and D on the graph, the function changes its nature from decreasing to
increasing or vice-versa These points may be called turning points of the given
function Further, observe that at turning points, the graph has either a little hill or a little
valley |
1 | 2806-2809 | Observe that at
points A, B, C and D on the graph, the function changes its nature from decreasing to
increasing or vice-versa These points may be called turning points of the given
function Further, observe that at turning points, the graph has either a little hill or a little
valley Roughly speaking, the function has minimum value in some neighbourhood
(interval) of each of the points A and C which are at the bottom of their respective
Rationalised 2023-24
APPLICATION OF DERIVATIVES
163
valleys |
1 | 2807-2810 | These points may be called turning points of the given
function Further, observe that at turning points, the graph has either a little hill or a little
valley Roughly speaking, the function has minimum value in some neighbourhood
(interval) of each of the points A and C which are at the bottom of their respective
Rationalised 2023-24
APPLICATION OF DERIVATIVES
163
valleys Similarly, the function has maximum value in some neighbourhood of points B
and D which are at the top of their respective hills |
1 | 2808-2811 | Further, observe that at turning points, the graph has either a little hill or a little
valley Roughly speaking, the function has minimum value in some neighbourhood
(interval) of each of the points A and C which are at the bottom of their respective
Rationalised 2023-24
APPLICATION OF DERIVATIVES
163
valleys Similarly, the function has maximum value in some neighbourhood of points B
and D which are at the top of their respective hills For this reason, the points A and C
may be regarded as points of local minimum value (or relative minimum value) and
points B and D may be regarded as points of local maximum value (or relative maximum
value) for the function |
1 | 2809-2812 | Roughly speaking, the function has minimum value in some neighbourhood
(interval) of each of the points A and C which are at the bottom of their respective
Rationalised 2023-24
APPLICATION OF DERIVATIVES
163
valleys Similarly, the function has maximum value in some neighbourhood of points B
and D which are at the top of their respective hills For this reason, the points A and C
may be regarded as points of local minimum value (or relative minimum value) and
points B and D may be regarded as points of local maximum value (or relative maximum
value) for the function The local maximum value and local minimum value of the
function are referred to as local maxima and local minima, respectively, of the function |
1 | 2810-2813 | Similarly, the function has maximum value in some neighbourhood of points B
and D which are at the top of their respective hills For this reason, the points A and C
may be regarded as points of local minimum value (or relative minimum value) and
points B and D may be regarded as points of local maximum value (or relative maximum
value) for the function The local maximum value and local minimum value of the
function are referred to as local maxima and local minima, respectively, of the function We now formally give the following definition
Definition 4 Let f be a real valued function and let c be an interior point in the domain
of f |
1 | 2811-2814 | For this reason, the points A and C
may be regarded as points of local minimum value (or relative minimum value) and
points B and D may be regarded as points of local maximum value (or relative maximum
value) for the function The local maximum value and local minimum value of the
function are referred to as local maxima and local minima, respectively, of the function We now formally give the following definition
Definition 4 Let f be a real valued function and let c be an interior point in the domain
of f Then
(a)
c is called a point of local maxima if there is an h > 0 such that
f (c) ≥ f (x), for all x in (c – h, c + h), x ≠ c
The value f (c) is called the local maximum value of f |
1 | 2812-2815 | The local maximum value and local minimum value of the
function are referred to as local maxima and local minima, respectively, of the function We now formally give the following definition
Definition 4 Let f be a real valued function and let c be an interior point in the domain
of f Then
(a)
c is called a point of local maxima if there is an h > 0 such that
f (c) ≥ f (x), for all x in (c – h, c + h), x ≠ c
The value f (c) is called the local maximum value of f (b)
c is called a point of local minima if there is an h > 0 such that
f (c) ≤ f (x), for all x in (c – h, c + h)
The value f (c) is called the local minimum value of f |
1 | 2813-2816 | We now formally give the following definition
Definition 4 Let f be a real valued function and let c be an interior point in the domain
of f Then
(a)
c is called a point of local maxima if there is an h > 0 such that
f (c) ≥ f (x), for all x in (c – h, c + h), x ≠ c
The value f (c) is called the local maximum value of f (b)
c is called a point of local minima if there is an h > 0 such that
f (c) ≤ f (x), for all x in (c – h, c + h)
The value f (c) is called the local minimum value of f Geometrically, the above definition states that if x = c is a point of local maxima of f,
then the graph of f around c will be as shown in Fig 6 |
1 | 2814-2817 | Then
(a)
c is called a point of local maxima if there is an h > 0 such that
f (c) ≥ f (x), for all x in (c – h, c + h), x ≠ c
The value f (c) is called the local maximum value of f (b)
c is called a point of local minima if there is an h > 0 such that
f (c) ≤ f (x), for all x in (c – h, c + h)
The value f (c) is called the local minimum value of f Geometrically, the above definition states that if x = c is a point of local maxima of f,
then the graph of f around c will be as shown in Fig 6 12(a) |
1 | 2815-2818 | (b)
c is called a point of local minima if there is an h > 0 such that
f (c) ≤ f (x), for all x in (c – h, c + h)
The value f (c) is called the local minimum value of f Geometrically, the above definition states that if x = c is a point of local maxima of f,
then the graph of f around c will be as shown in Fig 6 12(a) Note that the function f is
increasing (i |
1 | 2816-2819 | Geometrically, the above definition states that if x = c is a point of local maxima of f,
then the graph of f around c will be as shown in Fig 6 12(a) Note that the function f is
increasing (i e |
1 | 2817-2820 | 12(a) Note that the function f is
increasing (i e , f ′(x) > 0) in the interval (c – h, c) and decreasing (i |
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