Question

consider the following function.

$f(x)=4 - x^{2/3}$

find $f(-8)$ and $f(8)$.

$f(-8)=$

$f(8)=$

find all values $c$ in $(-8, 8)$ such that $f(c)=0$. (enter your answers as a comma - separated list. if an answer does not exist, enter dne.)

$c=$

based off of this information, what conclusions can be made about rolles theorem?

$\bigcirc$ this contradicts rolles theorem, since $f$ is differentiable, $f(-8)=f(8)$, and $f(c)=0$ exists, but $c$ is not in $(-8,8)$.

$\bigcirc$ this does not contradict rolles theorem, since $f(0)=0$, and 0 is in the interval $(-8,8)$.

$\bigcirc$ this contradicts rolles theorem, since $f(-8)=f(8)$, there should exist a number $c$ in $(-8,8)$ such that $f(c)=0$.

$\bigcirc$ this does not contradict rolles theorem, since $f(0)$ does not exist, and so $f$ is not differentiable on $(-8,8)$.

$\bigcirc$ nothing can be concluded.

Explanation:

Step1: Calculate \( f(-8) \)

Substitute \( x = -8 \) into \( f(x)=4 - x^{2/3} \).
\( f(-8)=4-(-8)^{2/3}=4 - [(-2)^3]^{2/3}=4 - (-2)^2=4 - 4=0 \)

Step2: Calculate \( f(8) \)

Substitute \( x = 8 \) into \( f(x)=4 - x^{2/3} \).
\( f(8)=4 - 8^{2/3}=4 - (2^3)^{2/3}=4 - 2^2=4 - 4=0 \)

Step3: Find the derivative \( f'(x) \)

Use the power rule \( (x^n)'=nx^{n - 1} \).
\( f'(x)=-\frac{2}{3}x^{-1/3}=-\frac{2}{3\sqrt[3]{x}} \)
Set \( f'(c) = 0 \), \( -\frac{2}{3\sqrt[3]{c}}=0 \). This equation has no solution for \( c\in(-8,8) \) (since the numerator is non - zero). Also, \( f(x) \) is not differentiable at \( x = 0 \) (because \( f'(x)=-\frac{2}{3\sqrt[3]{x}} \) has a vertical asymptote at \( x = 0 \)).

Rolle's Theorem requires that \( y = f(x) \) is continuous on \( [a,b] \), differentiable on \( (a,b) \), and \( f(a)=f(b) \). Here \( f(-8)=f(8) \), but \( f(x) \) is not differentiable on \( (-8,8) \) (due to non - differentiability at \( x = 0 \)).

Answer:

\( f(-8)=0 \)
\( f(8)=0 \)
\( c=\text{DNE} \)
The correct option is: This does not contradict Rolle's Theorem, since \( f'(0) \) does not exist, and so \( f \) is not differentiable on \( (-8,8) \).