Verify Rolle's theorem for the function $f(x) = x^2 + 2x - 8$, $x \in [-4, 2]$.

Let us check each condition of Rolle's theorem.

1) Function is continuous over [-4, 2]

2) Function is differentiable (-4, 2)


$f(-4) = (-4)^2 + 2(-4) - 8 = 16 -8 -8 = 0$

$f(2) = (2)^2 + 2(2) - 8 = 4 + 4 -8 = 0$

3) $f(-4) = f(2)$



$f(c) = c^2 + 2c - 8$

$f'(c) = 2c + 2 = 0$

4) $f'(c) = 0$

⇒ $2c + 2 = 0$ or $c = -1 \in (-4, 2)$