Question

Given the function `f(x)=x^3+3x^2-2`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-2,0] and find the c?

Answer 1

You determine whether it satisfies the hypotheses by determining whether `f(x) = x^3+3x^2-2` is continuous on the interval `[-2,0]` and differentiable on the interval `(-2,0)`.

You find the `c` mentioned in the conclusion of the theorem by solving `f'(x) = (f(0)-f(-2))/(0-(-2))` on the interval `(-2,0)`.

`f` is continuous on its domain, which includes `[-2,0]`

`f'(x) = 3x^2+6x` which exists for all `x` so it exists for all `x` in `(-2,0)`

Therefore this function satisfies the hypotheses of the Mean Value Theorem on this interval.

To find `c` solve the equation `f'(x) = (f(0)-f(-2))/(0-(-2))`. Discard any solutions outside `(-2,0)`.

I believe that you should get `c = (-3+-sqrt3)/3`. (Both are in `(-2,0)`.)