Rolle's Theorem

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) = x³ − x² − 6x + 5,    [0, 3]

Answer 1

Since f is a polynomial, it is continuous and differentiable. f(0)=f(3)=5. So f satisfies the three hypotheses of Rolle's Theorem. We now try to find all c such that f'(c)=0.

f'(x)=3x^2-2x-6. We now set f'(x)=0 and solve for x. Using the quadratic equation formula we get c=(sqrt(19)+1)/3