https://file.aiquickdraw.com/mathbot/file/1729746938126vqa6jadp.jpg

Let's go through each part of the problem step by step.

We are given the function \( f(x) = |x|^3 \), and we need to answer the following:

### (a) Find \( f'(x) \) for \( x \neq 0 \).

The absolute value function affects the expression of \( f(x) \) depending on whether \( x \) is positive or negative:

\[
f(x) = |x|^3 = 
\begin{cases} 
x^3 & \text{if } x \geq 0, \\
(-x)^3 = -x^3 & \text{if } x < 0.
\end{cases}
\]

Now, we differentiate each case:

- For \( x > 0 \), \( f(x) = x^3 \), so \( f'(x) = 3x^2 \).
- For \( x < 0 \), \( f(x) = -x^3 \), so \( f'(x) = -3x^2 \).

Thus, we can express the derivative as:

\[
f'(x) = 
\begin{cases} 
3x^2 & \text{if } x > 0, \\
-3x^2 & \text{if } x < 0.
\end{cases}
\]

### (b) Show that \( f(x) \) is differentiable at \( x = 0 \).

To check if \( f(x) \) is differentiable at \( x = 0 \), we first need to check if the derivative from both sides (left and right limits) at \( x = 0 \) exists and is equal.

- For \( x > 0 \), as \( x \to 0^+ \), we have \( f'(x) = 3x^2 \). So, \( \lim_{x \to 0^+} f'(x) = 0 \).
- For \( x < 0 \), as \( x \to 0^- \), we have \( f'(x) = -3x^2 \). So, \( \lim_{x \to 0^-} f'(x) = 0 \).

Both one-sided derivatives exist and are equal at \( x = 0 \). Therefore, \( f'(0) = 0 \).

Since \( f(x) \) is continuous at \( x = 0 \) (both left and right-hand limits of \( f(x) \) as \( x \to 0 \) equal \( 0 \)), and the derivative is continuous at \( x = 0 \), we conclude that \( f(x) \) is differentiable at \( x = 0 \).

### (c) Determine whether \( f'(x) \) is differentiable at \( x = 0 \).

We now need to check whether \( f'(x) \) is differentiable at \( x = 0 \). First, observe that:

\[
f'(x) = 
\begin{cases} 
3x^2 & \text{if } x > 0, \\
-3x^2 & \text{if } x < 0.
\end{cases}
\]

The derivative of \( f'(x) \) for \( x \neq 0 \) is:

\[
f''(x) = 
\begin{cases} 
6x & \text{if } x > 0, \\
-6x & \text{if } x < 0.
\end{cases}
\]

As \( x \to 0^+ \), \( f''(x) = 6x \to 0 \), and as \( x \to 0^- \), \( f''(x) = -6x \to 0 \). However, the left-hand and right-hand derivatives of \( f'(x) \) near \( x = 0 \) are:

- For \( x \to 0^+ \), the slope of \( f'(x) = 3x^2 \) approaches zero from the positive side.
- For \( x \to 0^- \), the slope of \( f'(x) = -3x^2 \) approaches zero from the negative side.

Thus, the left-hand and right-hand derivatives of \( f'(x) \) do not match near \( x = 0 \), indicating that \( f'(x) \) is **not differentiable at \( x = 0 \)**.

---

Would you like further details on any of these parts or have specific questions?

Here are 5 related questions for further practice:
1. What is the general definition of differentiability at a point?
2. How would the result change if the function was \( f(x) = |x|^n \) for some \( n > 3 \)?
3. Can you find an example of a function that is continuous but not differentiable at a point?
4. What are the conditions for a function to be continuously differentiable?
5. How does the concept of absolute value affect differentiability?

**Tip:** When checking differentiability, always check both the continuity of the function and the matching of the one-sided derivatives.

Let f(x) = |x|^3. (a) Find f'(x) for x ≠ 0. (b) Show that f(x) is differentiable at x = 0. (c) Determine whether f'(x) is differentiable at x = 0.

Solve the problem of differentiating the function f(x) = |x|^3, finding its derivative for x ≠ 0, and determining if f(x) and f'(x) are differentiable at x = 0. Key topics include absolute value functions and continuity.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation