Question

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.
$f(x)=8 - x^{2/3},-1,1$
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.

Explanation:

Step1: Check continuity

The function \( f(x)=8 - x^{\frac{2}{3}}\) is continuous on \([-1,1]\) since \(y = x^n\) is continuous for all real \(x\) when \(n>0\) and the sum of continuous functions is continuous.

Step2: Check differentiability

Differentiate \(f(x)\) using the power rule \( (x^n)^\prime=nx^{n - 1}\). So \(f^\prime(x)=-\frac{2}{3}x^{-\frac{1}{3}}=-\frac{2}{3\sqrt[3]{x}}\). The function \(f(x)\) is not differentiable at \(x = 0\) (because the derivative has a vertical asymptote at \(x=0\)).

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

Calculate \(f(-1)=8-(-1)^{\frac{2}{3}}=8 - 1=7\) and \(f(1)=8-(1)^{\frac{2}{3}}=8 - 1=7\).

Since the function \(f(x)\) is not differentiable on the open interval \((-1,1)\) (it is not differentiable at \(x = 0\in(-1,1)\)), Rolle's theorem does not apply.

Answer:

B. Rolle's Theorem does not apply.