Answer
$n \geq 16$
Work Step by Step
The error bound for Simpson's Rule is given as follows: $|E_S| \leq \dfrac{M(b-a)^5}{180n^4}$
Now, the maximum value of $|f^{4}(x)|$ on $[1,3]$ is: $|f^{4}(1)|=|\dfrac{24}{(1)^5}|=24$
$\implies M=24$
Further, $|E_S| \leq \dfrac{24 \times(3-1)^5}{180n^4}=\dfrac{32}{15n^4}$
we will choose $n$ such that $\dfrac{64}{15n^4} \leq 10^{-4}$
Therefore, $n \geq 14.4 \approx 15$
But as per Simpson's Rule, we need the even value of $n$ , thus $n \geq 16$ .
