Use Rolle's Theorem and argue the case that $\displaystyle f(x)=x^5-7x+c$ has at most one real root in the interval [-1,1]
You can show it is possible to have a root using the intermediate value theorem.
To use Rolle's theorem to argue there is at most one, you can proceed thusly:
Assume, to the contrary, there are two roots (or more, but at least 2 is fine), say for $\displaystyle \displaystyle x = x_1$ and $\displaystyle \displaystyle x = x_2$, both in the interval you are considering. and you may also assume that $\displaystyle x_1 < x_2$. then that means
$\displaystyle \displaystyle f(x_1) = f(x_2) = 0$
and so according to Rolle's theorem, there must be a point $\displaystyle \displaystyle x = x_3$, such that $\displaystyle \displaystyle x_1 < x_3 < x_2$ and $\displaystyle \displaystyle f'(x_3) = 0$.
Where can you get with that?