Hello to everyone,
I'm studying this function:

g(x) = m*f(x) - x

it seems pretty simple I know.
f(x) is a monotonically increasing and concave function
g(x)'s maximum in x will be dependent on m, obviously, call it x*:

g'(x) = m*f '(x) - 1 = 0

x* = inv_f '(1/m) (inverse of derivative function)

What I need is to show that f(x*) which means f(m) is increasing and concave in m, too. I tried with a couple of functions, for example f(x)=sqrt(x) and it worked. But how can I generalize that?

I'm looking forward to hearing a solution from you

Thank you and bye!