Verify that fx x3 2x 6 satisfies the hypothesis of the Me

Verify that f(x) = x3 - 2x + 6 satisfies the hypothesis of the Mean-Value Theorem over the interval [-2, 3] and find all values of C that satisfy the conclusion of the theorem.

Solution

f(x)=x^3-2x+6

f\'(x)=3x^2-2

f\'(c)=(f(y)-f(x))/(y-x)

3c^2-2=(-8+4+6)-(27-6+6)/(3-(-2)

3C^2-2=(2-27)/5

3C^2=-5+2

c^2=-3/3

c=±1