Answer
$n \geq 26$
Work Step by Step
The error bound for the Trapezoidal Rule is given as follows:
$|E_T| \leq \dfrac{M(b-a)^3}{12n^2}$
Now, the maximum value of $|f^{2}(x)|$ on $[0,1]$ is: $M=8$
Further, $|E_T| \leq \dfrac{8(1-0)^3}{12n^2}=\dfrac{2}{3n^4}$
we will choose $n$ such that $\dfrac{2}{3n^2} \leq 10^{-3}$
Therefore, $n \geq 25.8 \approx 26$
Thus $n \geq 26$
