# Thread: Rolle's Theorem and the Mean-value Theorem

1. ## Rolle's Theorem and the Mean-value Theorem

Show that hes exactly one root in .

2. Originally Posted by lacy1104
Show that hes exactly one root in .
On the interval $(0, 0.5\pi)$, the function $f(x) = x^3 + x^2 + 4x - cos x$ is negative at $x = 0$ but positive at $x = 0.5\pi$. If you can show that the derivative of f(x) is positive for $x \in (0, 0.5\pi)$, then you know that a root exists by the Intermediate Value Theorem, and that it is the only root by the Extreme Value Theorem.