Verify that f(x) = x^3 − 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.
f' = 3 x^2 -2
at -2 f(-2) = -8+4+6 = 2
at 3 f(3) = 27-6+6 = 27
mean slope = (27-2)/5 = 5
is f' = 5 in between -2 and +3?
5 = 3 x^2-2
3 x^2 = 7
x^2 = 7/3
x = - 1.53 and +1.53
both are between x = -2 and x = +3