real analysis – Shortest path and convexity


I found a PDF in which there were the two following questions:

  1. For any function $g in C^1$, let $F(g) = int_{0}^{1} sqrt{1 + (f’(t))^2}dt$. Show that if f, g are $C^1$ and $lambda in )0, 1($, then $F(lambda f + (1 – lambda)g) leq lambda F(f) + (1-lambda) F(g)$.

  2. If $a, b$ are given in $mathbb{R}$, show that there is a function f with $f(0)=a, f(1)=b$ such that for each function g with $g(0)=a, g(1)=b$, we have $F(f) leq F(g)$.

I’m good with question 1) (I used the convexity of $x mapsto sqrt{1+x^2}$).

I get the meaning of question 2 (basically, the goal is to show that amongst all $C^1$ functions, then it’s the linear function satisfying the conditions that is the shortest). However, i can’t really see how to use question 1 to answer question 2 …

Thank you.