I was wondering if someone could help me with the following problem. To be honest, I don't really know where to start ...
A profit maximising firm producing output using a single input according to a production function $f(\cdot):\mathbb{R}_+ \to \mathbb{R}_+$ will face an optimisation problem of the following sort
maximise $pf(x)-wx$ over $x\in \mathbb{R}_+$
where $w$ is the unit price of the input and $p$ is the unit price of the output.
Show that if the production function $f$ is strictly concave then given any $p\geq 0$ and $w>0$ the firm's optimisation problem has a unique solution in $\mathbb{R}_+$.