If production function is concave, then demonstrate that profit function will also be concave Show that concavity of firm's production function implies concavity of its profit function.
(Hint: For a concave function, first order conditions gives the vector that maximizes the function)
Confusing point: How do I relate the concavity of production function to the profit function. And where do I begin this proof mathematically.
The &quot;hint&quot; is wrong and misleading.
It is wrong, because first-order conditions are sufficient for a maximum when the objective function is strictly concave, or at least strictly-quasi-concave, under some conditions. For example, a linear function is also concave (as well as convex), and in such a case, the first-order conditions are not sufficient for a maximum.
It is misleading, because concavity of a multivariate function depends on the signs of its Hessian matrix of second-order derivatives.  So the OP should compute the second derivatives of the profit function, with respect to capital and labor, since I guess the presumption here is that we maximize with respect to input quantities only, treating prices as exogenous constants.
Both of the above are standard material in many microeconomics books or &quot;math for economics&quot; books, so the OP should look them up there.
The proper &quot;hint&quot; would be that costs are linear in capital and labor...
...so the profit function has a non-linear part that relates to production function, and a linear part that relates to costs. And what happens to the second derivative when linearity is present ?