Maximizer of a function varying smoothly depending on a parameter

Question

Let $f:\mathbb{R}^2\to \mathbb{R}$ be a smooth function such that for all $t$ $f_t:=f(t,\cdot)$ has a unique maximum at $x^*(t)$ (e.g. $f_t$ is a strictly concave function for all $t$). My question is: is the function $t\mapsto x^*(t)$ smooth in general? If not, are there some reasonable conditions on $f$ that guarantee its smoothness?

Answer 1

Thinking about it a bit more I realized that under the assumption of strict concavity the answers follows simply from the implicit function theorem since $\frac{\partial f}{\partial x}(t,x^*(t))=0$ and $\frac{\partial^2 f}{\partial x^2}(t,x^*(t))<0$.