Question

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.) f(x) = x^3 − x^2 − 12x + 3,    [0, 4]

Answer 1

Well, is the function continuous?  yes, is a cubic one, you can graph it if you wish, is continuous all the way, and of course at [ 0, 4] too.

is it differentiable?  you can always look at the graph between 0,4 and is a smooth transition line, thus yes, it is differentiable, but let's check anyway,

$\bf \cfrac{dy}{dx}=3x^2-2x-12$    its derivative has no asymptotes and therefore no "cusps", so yes, is differentiable all around.

is f(0) = f(4), let's check

f(0) = 0+0+0+3,            f(0) = 3
f(4) = 64 - 16 - 48 + 3,   f(4) = 3

yeap

there must then be a "c" value(s) with a horizontal tangent slope, let's check, is really just the critical points.

$\bf \cfrac{dy}{dx}=3x^2-2x-12\implies 0=3x^2-2x-12 \\\\\\ \textit{using the quadratic formula} \\\\\\ x=\cfrac{-(-2)\pm\sqrt{(-2)^2-4(3)(-12)}}{2(3)}\implies x=\cfrac{2\pm\sqrt{4+144}}{6} \\\\\\ x=\cfrac{2\pm 2\sqrt{37}}{6}\implies x=\cfrac{2\pm\sqrt{37}}{3}\impliedby \textit{c's}$