A question about convex functions of minimum and maximum

Question

A question about convex functions of minimum and maximum

Let $f_1, f_2$ are two convex functions $\mathbb{R^n}\to \mathbb{R}$

then I prove that their $\max\{f_1(x),f_2(x),\forall x\in \mathbb{R}^n$ is convex

Is their minimum is also convex?

Answer 1

take $f_1(x)=x^2$ and $f_2(x)=x^2+7$

then $ f(x)=\min \{f_1(x),f_2(x)\}$ is not convex function

Answer 2

$f(x)=x$ and $g(x)=-x$ gives a counterexample.