Let $f : \mathbb{R} \longrightarrow \mathbb{R}$ be a differentiable function such as $\forall x \in \mathbb{R}, f'(x) > 2$.
What could I do to show that $\exists c \in \mathbb{R} : f(c) = 0$ ?
At first I thought I could use the mean value theorem but it did not help me. In fact, I don't really understand why $f(x)$ couldn't be strictly positive $\forall x \in \mathbb{R}$.
Would you know how to proceed ?
Thanks for your help.
Best Answer
From the mean-value theorem you get $$ \frac{f(x)-f(0)}{x-0} > 2 $$ for all nonzero $x$. It follows that $$ f(x) \ge f(0) + 2x \text{ for } x > 0 $$ and $$ f(x) \le f(0) + 2x \text{ for } x < 0 $$ so that $f$ necessarily takes both positive and negative values. Now conclude with the intermediate value theorem.