How to show the production function is concave in K and L but not strictly so?
So we want the Hessian to be NSD, so we need the PMs to alternate weakly. 

$H=\begin{bmatrix} F_{kk} &amp; F_{kl} \\
F_{kl} &amp; F_{ll}
\end{bmatrix}~~NSD \iff~~~F_{kk},F_{ll}\leq0~~~\&amp;~~F_{kk}F_{ll}-F_{kl}^2\geq0$

We are given that $F_{kk},F_{ll}&lt;0$ so we need to figure out the cross partials.

Constant returns to scale implies that we have a homogeneous of degree 1 function: 

$F(K,L)=KF_k+LF_l \implies  F_l=KF_{kl}+LF_{ll}+F_l~~ \&amp;~~  F_k=LF_{kl}+KF_{kk}+F_k$

Now we can substitute these equations into the expression for the 2nd PM:

$\big(-\frac{L}{K}F_{kl}\big)\big(-\frac{K}{L}F_{kl}\big)-F_{kl}^2=0~~~~~$
 and we're done.