Help show that a second derivative is always negative

Question

How do I show that the second derivative is always negative?

I've computed the second derivative to be:

$\displaystyle\frac{n}{2\sigma^4}-\frac{1}{\sigma^6}\sum\limits_{i=1}^n(x_i-\mu)^2$

Then I don't know what to do next, mainly because I don't know how to deal with the summation in the second term.

Also, if $\mu$ is unknown, then $\mu= \bar{x}$. How will that change the answer?

Note:

$n>0$

$X \sim N(\mu,\sigma^2)$

Answer 1

There is an equation in the material you provided that tells us that the summation term is equal to $n\sigma^2$. I expect you can take it from there.