Optimization for minimum cost, with the total cost function $C=TQ^{1/a} + F$ I have the function $C=TQ^{1/a} + F$. Where C is total cost, Q is output, a is a positive parametric constant, F is fixed cost, and T measures the technology available to the firm (Parameter). We also know that $T > 0$.
A hint is to use calculus of optimization, and to check our second-order condition.
I got the answer of it being convex with the function of $T(1/a)(1/a-1)Q^{1/a-2}$
However I can't figure out the rest:
Assuming $a<0.5$, and by using optimization what is the value of q that minimizes the average cost?
Hint: the answer for $q$ will be in the parameters of the model, be sure to check the second order condition.

$$AC=\frac{C}{Q}=TQ^{1/a-1}+FQ^{-1}$$
$$AC'=T(1/a-1)Q^{1/a-2}-FQ^{-2}=0 \Rightarrow Q=\left(\frac{F}{T(1/a-1)}\right)^a$$
$$AC''=T(1/a-1)(1/a-2)Q^{1/a-3}+2FQ^{-3}>0,$$
because $a<0.5$.
