Question

How do you check the hypotheses of Rolle's 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

Hint : As we know that Rolle's theorem is an extension or special case of mean value theorem and therefore it is interrelated. There is a function that satisfies like $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$.
While in the mean value theorem says that in a particular that if $f$ is a function and $a,b$ are two endpoints then there has to be appoint where the tangent curve of the slope is equal to the slope of secants then it has the closed intervals $[a,b]$ and differentiable at open intervals $(a,b)$.

Complete step by step solution:
In the above question we have a function $f(x) = {x^3} + {x^2}$.
From the above definitions we can see that 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,
But we can see that we have no interval given. Without knowing the interval we can not test the third hypotheses for Rolle's theorem nor we can state the conclusions for the theorems.
Hence we can not check the hypotheses of Rolle's theorem and mean value theorem.

Note : Before solving this kind of question we should have the clear idea what the definition means of Rolle's theorem and mean value theorem. We should know that Rolle's theorem is a special variant of the Lagrange’s mean value theorem or we can say the extension of the primary concept.