determine whether rolles theorem applies to the function shown below on the given interval. if so, find the…

determine whether rolles theorem applies to the function shown below on the given interval. if so, find the point(s) that are guaranteed to exist by rolles theorem. f(x)=6 - x^(2/3),-1,1 select the correct choice and, if necessary, fill in the answer box to complete your choice. a. rolles theorem applies and the point(s) guaranteed to exist is/are x = (type an exact answer, using radicals as needed. use a comma to separate answers as needed.) b. rolles theorem does not apply.

determine whether rolles theorem applies to the function shown below on the given interval. if so, find the point(s) that are guaranteed to exist by rolles theorem. f(x)=6 - x^(2/3),-1,1 select the correct choice and, if necessary, fill in the answer box to complete your choice. a. rolles theorem applies and the point(s) guaranteed to exist is/are x = (type an exact answer, using radicals as needed. use a comma to separate answers as needed.) b. rolles theorem does not apply.

Answer

Explanation:

Step1: Check continuity

The function $f(x)=6 - x^{\frac{2}{3}}$ is continuous on the closed - interval $[-1,1]$ since $y = x^{\frac{2}{3}}=\sqrt[3]{x^{2}}$ is a composition of a polynomial and a cube - root function, and $f(x)$ is a linear combination of a constant and $x^{\frac{2}{3}}$.

Step2: Check differentiability

Find the derivative of $f(x)$ using the power rule. The derivative $f^\prime(x)=-\frac{2}{3}x^{-\frac{1}{3}}=-\frac{2}{3\sqrt[3]{x}}$. The function is not differentiable at $x = 0$ which is in the open interval $(-1,1)$.

Step3: Check $f(-1)$ and $f(1)$

Calculate $f(-1)=6-(-1)^{\frac{2}{3}}=6 - 1=5$ and $f(1)=6-(1)^{\frac{2}{3}}=6 - 1=5$. Since $f(-1)=f(1)$, but the function is not differentiable on the open interval $(-1,1)$ (due to non - differentiability at $x = 0$), Rolle's theorem does not apply.

Answer:

B. Rolle's Theorem does not apply.