How can I apply Rolle's theorem here?

Question

How can I use Rolle's theorem to check whether the function $f(x)=x^7+x−7$ has two real roots ?

Answer 1

By Rolle's theorem, if there are two roots, there must be a stationary point, i.e. we need a $c$ for which $f'(c) = 0$ (this $c$ must even be between the roots, but that is of little matter here). But $f'(x) = 7x^6 + 1 \geq 1$ can never be $0$. Therefore there can be no such $c$ and therefore there can be no second root of $f$.