Given the function f(x)=x^3+3x^2-2f(x)=x3+3x22, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-2,0] and find the c?

1 Answer
Dec 4, 2016

You determine whether it satisfies the hypotheses by determining whether f(x) = x^3+3x^2-2f(x)=x3+3x22 is continuous on the interval [-2,0][2,0] and differentiable on the interval (-2,0)(2,0).

You find the cc 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).)