Hi all ----

I actually get part (ii) of this question - but I'm just curious - how can I formally prove the symmetry? The question doesn't ask this but I'm just curious. The green box is the solution.

Part (ii) ---I can see that $\displaystyle \int_0^1 2^{-(x - c)^2}$ reaches maximum when $\displaystyle c = 1/2$.

But it also looks like $\displaystyle 2^{-\left(x - \frac{1}{2})^2\right}$ is symmetric about $\displaystyle x = 1/2$. How would I prove this?

I know how to do this for $\displaystyle f(x) = 2^{-x^2}$ ---

$\displaystyle f(x)$ is symmetric about $\displaystyle x = 0$ because $\displaystyle f(-x) = 2^{-(-x)^2} = f(x) = 2^{-x^2} $ so $\displaystyle f(-x) = f(x)$.

Thanks a lot ---