On anti-derivative of functions

Question

Let $f,g,h: \mathbb R \to \mathbb R$ be differentiable functions.

(1) Does there necessarily exist a differentiable function $F: \mathbb R \to \mathbb R $ such that $F'=\max \{f' ,g' \}$ ?

(2) Does there necessarily exist a differentiable function $G: \mathbb R \to \mathbb R $ such that $G '=\max \{f',g',h'\}$ ?

Answer 1

Not an answer to the actual question, but note that (1) and (2) are equivalent: (2) follows from applying (1) twice since $\max\{f',g',h'\}=\max\{\max\{f',g'\},h'\}$ and (1) follows from (2) by taking $h=g$.