Max of Second Derivative, Wrong answer apparently

Question

I am given the integral: $\int_{0}^{4}5e^{-x^{2}}dx$. I am told to find the Error bound using the Midpoint rule, which is $\frac{K(b-a)^3}{24n^2}$. After finding the second derivative of the given formula, I used my calculator to find that the max of the second derivative is $4.4626032$. I have plugged in this answer into my webwork question, but it has marked it wrong for some reason, despite the fact that this is the only maximum point within the interval $[0,4]$. What am I doing wrong?

Answer 1

Let $f (x)=5e^{-x^2} $. then

$$f'(x)=-10xe^{-x^2} $$

$$f''(x)=(-10+20x^2)e^{-x^2} $$

$$f'''(x)=20x (3-2x^2)e^{-x^2} $$

$$\max_{0\le x\le 4} f''(x)=f''(\frac {\sqrt {3}}{\sqrt {2}}) $$ $$=20e^{-\frac {3}{2}} $$ $$\approx 4.4626$$