# 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.