Consider the following function.
f(x) = 4 − x2/3
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 Rolle's Theorem?

This contradicts Rolle's Theorem, since f is differentiable, f(−8) = f(8), and f '(c) = 0 exists, but c is not in (−8, 8).
This does not contradict Rolle's Theorem, since f '(0) = 0, and 0 is in the interval (−8, 8).    
This contradicts Rolle's Theorem, since f(−8) = f(8), there should exist a number c in (−8, 8) such that f '(c) = 0.
This does not contradict Rolle's Theorem, since f '(0) does not exist, and so f is not differentiable on (−8, 8).
Nothing can be concluded.

Let's solve this step-by-step.

### Step 1: Evaluate \( f(-8) \) and \( f(8) \)
The given function is:

\[
f(x) = 4 - x^{2/3}
\]

Now, substitute \( x = -8 \) and \( x = 8 \) to find \( f(-8) \) and \( f(8) \).

#### For \( f(-8) \):
\[
f(-8) = 4 - (-8)^{2/3}
\]

The cube root of \( -8 \) is \( -2 \), and squaring \( -2 \) gives \( 4 \). So,

\[
f(-8) = 4 - 4 = 0
\]

#### For \( f(8) \):
\[
f(8) = 4 - (8)^{2/3}
\]

The cube root of \( 8 \) is \( 2 \), and squaring \( 2 \) gives \( 4 \). So,

\[
f(8) = 4 - 4 = 0
\]

Thus, \( f(-8) = 0 \) and \( f(8) = 0 \).

### Step 2: Find \( f'(x) \)
Now, we differentiate the function \( f(x) = 4 - x^{2/3} \).

Using the power rule for differentiation:

\[
f'(x) = \frac{d}{dx}\left(4 - x^{2/3}\right) = 0 - \frac{2}{3}x^{-1/3} = -\frac{2}{3}x^{-1/3}
\]

So,

\[
f'(x) = -\frac{2}{3} \cdot \frac{1}{\sqrt[3]{x}}
\]

### Step 3: Find \( c \) such that \( f'(c) = 0 \)
Set \( f'(x) = 0 \):

\[
-\frac{2}{3} \cdot \frac{1}{\sqrt[3]{x}} = 0
\]

For this equation to hold, there is no value of \( x \) that makes \( -\frac{2}{3} \cdot \frac{1}{\sqrt[3]{x}} = 0 \). This implies that **there is no \( c \) in the interval** \( (-8, 8) \) such that \( f'(c) = 0 \).

Therefore, the answer for \( c \) is:

\[
c = \text{DNE (Does Not Exist)}
\]

### Step 4: Rolle's Theorem
Rolle's Theorem states that if a function \( f \) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \in (a, b) \) such that \( f'(c) = 0 \).

In this case:
- \( f(-8) = f(8) = 0 \), so the first condition is satisfied.
- However, the derivative \( f'(x) \) involves \( x^{-1/3} \), which is undefined at \( x = 0 \). This means the function is **not differentiable** at \( x = 0 \).

Thus, **this does not contradict Rolle's Theorem**, since the function is not differentiable on the entire interval \( (-8, 8) \).

The correct conclusion is:
\[
\boxed{\text{This does not contradict Rolle's Theorem, since } f'(0) \text{ does not exist, and so } f \text{ is not differentiable on } (-8, 8).}
\]

---

Do you need more details on any part of this explanation?

Here are some related questions you could explore:
1. How is the cube root of a negative number defined in general?
2. What are the key conditions of Rolle's Theorem?
3. What happens if a function is not differentiable at a single point in the interval?
4. Can a function still be continuous even if it's not differentiable everywhere?
5. How do we find critical points where the derivative doesn't exist?

**Tip:** Always check if the function is differentiable across the entire interval when applying Rolle's Theorem!

Consider the following function. f(x) = 4 − x^(2/3). Find f(−8) and f(8). Find all values c in (−8, 8) such that f '(c) = 0. Based off of this information, what conclusions can be made about Rolle's Theorem?

Explore how Rolle's Theorem applies to the function f(x) = 4 - x^(2/3), where differentiability and critical points are key to solving. This detailed breakdown covers finding f(−8), f(8), and evaluating the existence of a critical value c in (−8, 8) based on Rolle's Theorem.

Math Problem Statement

f(−8)

f(8)

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation