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!