I understood the question but I got it Wrong...

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 − 2x + 8,    [0, 2]

As far as Rolles Therem, I know it is supposed to equal c, but when I plugged it in, I dot 3x^2 x 2x = 2, therefore making the zeroes 2, and 2/3. I still got wrong answer though...

Answer 1

You first show that it satisfies the three requirements: f(a) = f(b), f is continuous on [a,b], and f is differentiable on (a,b).

As f(x) = x^3 - x^2 -2x +8, f(x) is a polynomial, so it is continuous and differentiable everywhere.

f(0) = 8

f(2) = 2^3 - 2^2 - 2*2 + 8 = 8 - 4 - 4 + 8 = 8

Thus, as f(0) = 8 and f(2) = 8, f(0) = f(2)

Michael S, was correct about there being one critical value in the interval.

From the quadratic formula, the zeroes are at (1 +- sqrt(7))/3, and the value in the interval is at (1 + sqrt(7))/3, which is approximately 1.22

Answer 2

I am unsure how you determined that the critical points would be at 2 and 2/3.

The derivative of f(x) = x³ − x² − 2x + 8 is

f '(x) = 3x² − 2x − 2

Using the quadratic formula you find the only positive critical point at 1.22

This is acceptable for the Rolle's Theorum since1.22 is between 0 and 2.

There must have been some calculator error.