Question

How do you check the hypotheses of Rolles Theorem and the Mean Value Theorem and find a value of c that makes the appropriate conclusion true for `f(x) = x^3+x^2`?

Answer 1

The hypotheses and conclusions for both of those theorems involve a function on an interval.

The function `f(x)=x^3+x^2` is a polynomial, so it is continuous on any closed interval and differentiable on the interior of any closed interval.

Without knowing the interval we cannot test the third hypothesis for Rolle's Theorem, nor can we state the conclusions for the theorems.