Mean Value Theorem for Convex derivatives

Question

Let $f:\mathbb R^n\to\mathbb R^m$ be differentiable. Assume that the set of derivatives $$\{f'(x)\in L(\mathbb R^n,\mathbb R^m):x\in [a,b]\} \text{ is convex.}$$

Prove that there exist a $θ$ in $[a,b]$ such that $f(b)−f(a)=f'(θ)(b−a)$.

This is problem number $17$ from chapter $5$ of "Real mathematical analysis" by Charles Chapman Pugh [Springer (February 19, 2010)]. If $f$ is $C^1$, there is a straightforward proof. Could anyone help to prove or disprove this statement without $C^1$ condition?

Answer 1

This is only a partial answer, since it assumes that $f$ is $C^1$.

Here are some hints if $f$ is $C^1$.

Let $R = \{ f'(x)(b-a) | x \in [a,b] \}$ and note that $R$ is convex and compact (since $f'([a,b])$ is compact).

Note that $p \in R$ iff $\phi^T p \in \phi^T R$ for all linear functionals $\phi$.

You want to show that $f(b)-f(a) \in R$.

Note that the ordinary mean value theorem applies to $\phi^T f$.