Rolle's theorem question to show there exists a $\space c \space$ s.t. $\space f'(c)=2c$.

Question

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function satisfying that $f(0)=0$ and $f(1)=1$. Using Rolle's theorem show that there exists $c \in (0,1)$ such that $f'(c)=2c$.

Answer 1

HINT

Consider $g(x) = f(x) - x^2$ and note that $g(0)=0$ and $g(1) = 1-1^2=0$. Apply Rolle's Theorem to $g(x)$.