Minimize $F[u(\cdot)]=\frac{1}{2}\int_0^1f(u'(x))dx$

Question

Let $C^2\ni f:\mathbb{R}\rightarrow\mathbb{R}$ be a strictly convex function and $F$ be the functional defined by $$F[u(\cdot)]:= \frac{1}{2}\int_0^1f(u'(x))dx,$$ acting on functions in the set $\mathscr{A}:= \{u:[0,1]\rightarrow\mathbb{R}:u\in C^1, u(0)=A,u(1)=B\}$.

What is the minimizer of $F$ on $\mathscr{A}?$

I've reduced the problem to $f'(u'(x))=2a$ where $a\in\mathrm{R}$ using Euler-Lagrange. However, I don't know how to solve this differential any further. I have a feeling I should be using the fact that $f$ is strictly convex, but I don't know how it would help.

Any advice on how to proceed would be appreciated! Please don't spoil the answer :)

Answer 1

Since $f$ is strictly convex, $f^\prime$ is strictly... what?

What does that tell you about $u^\prime$? What does that tell you about $u$?