Given the values:
| \(x\) |
\(f(x)\) |
| 5 |
150 |
| 7 |
392 |
| 11 |
1452 |
| 13 |
2366 |
| 17 |
5202 |
We need to evaluate \( f(9) \) using Newton’s divided difference formula.
First, we compute the divided differences:
-
First divided differences:
\[
\begin{aligned}
f[5, 7] &= \frac{392 - 150}{7 - 5} = 121, \\
f[7, 11] &= \frac{1452 - 392}{11 - 7} = 265, \\
f[11, 13] &= \frac{2366 - 1452}{13 - 11} = 457, \\
f[13, 17] &= \frac{5202 - 2366}{17 - 13} = 709.
\end{aligned}
\]
-
Second divided differences:
\[
\begin{aligned}
f[5, 7, 11] &= \frac{265 - 121}{11 - 5} = 24, \\
f[7, 11, 13] &= \frac{457 - 265}{13 - 7} = 32, \\
f[11, 13, 17] &= \frac{709 - 457}{17 - 11} = 42.
\end{aligned}
\]
-
Third divided differences:
\[
\begin{aligned}
f[5, 7, 11, 13] &= \frac{32 - 24}{13 - 5} = 1, \\
f[7, 11, 13, 17] &= \frac{42 - 32}{17 - 7} = 1.
\end{aligned}
\]
-
Fourth divided difference:
\[
f[5, 7, 11, 13, 17] &= \frac{1 - 1}{17 - 5} = 0.
\]
Using these divided differences, the interpolating polynomial is constructed as:
\[
P(x) = 150 + 121(x - 5) + 24(x - 5)(x - 7) + 1(x - 5)(x - 7)(x - 11)
\]
Evaluating this polynomial at \( x = 9 \):
\[
\begin{aligned}
P(9) &= 150 + 121(9 - 5) + 24(9 - 5)(9 - 7) + 1(9 - 5)(9 - 7)(9 - 11) \\
&= 150 + 121 \cdot 4 + 24 \cdot 4 \cdot 2 + 1 \cdot 4 \cdot 2 \cdot (-2) \\
&= 150 + 484 + 192 - 16 \\
&= 810.
\end{aligned}
\]
Thus, the value of \( f(9) \) is:
\[
\boxed{810}
\]