Say I have a functional of the form,
$$ F(x, f(x); c) = \max_{f(x),c} \left\{ f(x)c - 1/2 c^2 \right\} $$
with $c$ a parameter. Can I take first-order conditions with respect to c and substitute $c^* = f(x)$ back into the functional and still get the same solution for $f^*(x)$?
If only so under certain conditions, which conditions would suffice?