Proof of Concavity of Cobb-Douglas Function
If you look at the case $x_1=x_2=\ldots x_n=t&gt;0$, you have
$$F(x_1,x_2,\ldots,x_n)=t^{a_1}\cdot t^{a_2}\cdots t^{a_n}.$$
Verify that the function $t\mapsto t^{a_1}\cdot t^{a_2}\cdots t^{a_n}$ is not concave. This implies that $F$ is not concave either.