I am given a function $f(x) = x^3 + 3x - 1$, and I am asked to prove that $f(x)$ has exactly one real root using the Intermediate Value Theorem and Rolle's theorem. So far, I managed to prove the existence of at least one real root using IVT.
Note that $f(x)$ is continuous and differentiable for all $x \in \Bbb R$. By inspection, since $f(-1) = -5$ and $f(1) = 3$, by the IVT, there exists a $c \in [-1, 1] $ such that $f(c) = 0$. Therefore, $f(x)$ must have at least one root. I am not sure if this is the best way to prove that it has at least one root - if there is a better way, please let me know. I don't know how I can use Rolle's theorem to prove that there is only one root. I had an idea of integrating the polynomial and then proving that it has just one maximum using Rolle's theorem but that did not work out so well.