Q:

find the maximum

I'm stuck in this problem (Instructions and my work is showed on the picture below) I just don't know what to do right after find the values of D, to prove that the maximum of F(D) is attained at the smaller of the two positive solution of the equation. Please see my work in the picture below

A:

Edit
First of all, we need to make sure that $\alpha$ is negative such that the function $f(D) = \alpha D^2 + \beta D + M_o^2$ opens downward.
If yes, then a maximum is attained at $D = - \frac {\beta }{2\alpha}$ and the maximum value is given by $f(\frac {-\beta}{2\alpha})$.

