# Questions about profit function [duplicate]

I know that a profit function is defined by $$\pi(p;w)=\max_{x\in\mathbb{R}}pf(x)-w\cdot x$$ and $$\pi$$ is a convex function (i.e. $$\pi$$ is convex in $$(p, w)$$. My concern is how we can find the max value of the profit function when the profit function itself is convex.

From my previous knowledge on convex function, I also know that a convex function has a global minimum on its domain (correct me if I was wrong), but how can we find the maximum?

Can anyone please explain this for me? I am so confused. Thanks in advanced!

You're probably confusing two things: the objective function $$pf(x) - wx$$ being used in profit maximization problem: $$\max pf(x) - wx$$, the objective function is concave in $$x$$ and thus, you can solve the maximization problem. However, a profit function $$\pi(p,w)$$ being a result of maximization problem identifies the maximum profit given level of prices and wages $$\pi^*(p,w)$$ (and it's convex in $$p$$ and $$w$$ as you said).