To determine which of the given functions is thrice differentiable at \( x = 0 \), we will analyze each function step by step.
### Given Functions:
1. \( f(x) = |x^3| \)
2. \( f(x) = x^3 |x| \)
3. \( f(x) = x |x| \sin^3(x) \)
4. \( f(x) = x \tan^3(x) \)
### Step 1: Analyze \( f(x) = |x^3| \)
- For \( x \geq 0 \): \( f(x) = x^3 \)
- For \( x < 0 \): \( f(x) = -x^3 \)
**Check Continuity at \( x = 0 \):**
\[
\lim_{x \to 0^+} f(x) = 0^3 = 0
\]
\[
\lim_{x \to 0^-} f(x) = -(-0^3) = 0
\]
Since both limits equal \( f(0) = 0 \), \( f(x) \) is continuous at \( x = 0 \).
**Check Differentiability:**
- For \( x > 0 \): \( f'(x) = 3x^2 \)
- For \( x < 0 \): \( f'(x) = -3x^2 \)
**At \( x = 0 \):**
\[
\lim_{x \to 0^+} f'(x) = 3(0)^2 = 0
\]
\[
\lim_{x \to 0^-} f'(x) = -3(0)^2 = 0
\]
Thus, \( f'(0) = 0 \), and \( f'(x) \) is continuous at \( x = 0 \).
**Second Derivative:**
- For \( x > 0 \): \( f''(x) = 6x \)
- For \( x < 0 \): \( f''(x) = -6x \)
**At \( x = 0 \):**
\[
\lim_{x \to 0^+} f''(x) = 6(0) = 0
\]
\[
\lim_{x \to 0^-} f''(x) = -6(0) = 0
\]
Thus, \( f''(0) = 0 \), and \( f''(x) \) is continuous at \( x = 0 \).
**Third Derivative:**
- For \( x > 0 \): \( f'''(x) = 6 \)
- For \( x < 0 \): \( f'''(x) = -6 \)
**At \( x = 0 \):**
\[
\lim_{x \to 0^+} f'''(x) = 6
\]
\[
\lim_{x \to 0^-} f'''(x) = -6
\]
Since the limits do not match, \( f'''(0) \) does not exist. Therefore, \( f(x) = |x^3| \) is **not** thrice differentiable at \( x = 0 \).
### Step 2: Analyze \( f(x) = x^3 |x| \)
- For \( x \geq 0 \): \( f(x) = x^4 \)
- For \( x < 0 \): \( f(x) = -x^4 \)
**Check Continuity at \( x = 0 \):**
\[
\lim_{x \to 0^+} f(x) = 0^4 = 0
\]
\[
\lim_{x \to 0^-} f(x) = -(-0^4) = 0
\]
Thus, \( f(x) \) is continuous at \( x = 0 \).
**Check Differentiability:**
- For \( x > 0 \): \( f'(x) = 4x^3 \)
- For \( x < 0 \): \( f'(x) = -4x^3 \)
**At \( x = 0 \):**
\[
\lim_{x \to 0^+} f'(x) = 4(0)^3 = 0
\]
\[
\lim_{x \to 0^-} f'(x) = -4(0)^3 = 0
\]
Thus, \( f'(0) = 0 \), and \( f'(x) \) is continuous at \( x = 0 \).
**Second Derivative:**
- For \( x > 0 \): \( f''(x) = 12x^2 \)
- For \( x < 0 \): \( f''(x) = -12x^2 \)
**At \( x = 0 \):**
\[
\lim_{x \to 0^+} f''(x) = 12(0)^2 = 0
\]
\[
\lim_{x \to 0^-} f''(x) = -12(0)^2 = 0
\]
Thus, \( f''(0) = 0 \), and \( f''(x) \) is continuous at \( x = 0 \).
**Third Derivative:**
- For \( x > 0 \): \( f'''(x) = 24x \)
- For \( x < 0 \): \( f'''(x) = -24x \)
**At \( x = 0 \):**
\[
\lim_{x \to 0^+} f'''(x) = 24(0) = 0
\]
\[
\lim_{x \to 0^-} f'''(x) = -24(0) = 0
\]
Thus, \( f'''(0) = 0 \), and \( f(x) = x^3 |x| \) is thrice differentiable at \( x = 0 \).
### Step 3: Analyze \( f(x) = x |x| \sin^3(x) \)
- For \( x \geq 0 \): \( f(x) = x^2 \sin^3(x) \)
- For \( x < 0 \): \( f(x) = -x^2 \sin^3(x) \)
**Check Continuity at \( x = 0 \):**
\[
\lim_{x \to 0^+} f(x) = 0^2 \sin^3(0) = 0
\]
\[
\lim_{x \to 0^-} f(x) = -(-0^2 \sin^3(0)) = 0
\]
Thus, \( f(x) \) is continuous at \( x = 0 \).
**Check Differentiability:**
- For \( x > 0 \): \( f'(x) = 2x \sin^3(x) + 3x^2 \sin^2(x) \cos(x) \)
- For \( x < 0 \): \( f'(x) = -2x \sin^3(x) - 3x^2 \sin^2(x) \cos(x) \)
**At \( x = 0 \):**
Both derivatives approach \( 0 \) as \( x \to 0 \). Thus, \( f'(0) = 0 \).
**Second Derivative:**
This will involve product and chain rules and will also yield \( f''(0) = 0 \).
**Third Derivative:**
This will also yield \( f'''(0) = 0 \).
### Step 4: Analyze \( f(x) = x \tan^3(x) \)
- For \( x \geq 0 \): \( f(x) = x \tan^3(x) \)
- For \( x < 0 \): \( f(x) = -x \tan^3(-x) = -x (-\tan^3(x)) = x \tan^3(x) \)
**Check Continuity at \( x = 0 \):**
\[
\lim_{x \to 0} f(x) = 0
\]
Thus, \( f(x) \) is continuous at \( x = 0 \).
**Check Differentiability:**
Using the product rule and chain rule, we can find that \( f'(0) = 0 \).
**Second Derivative:**
Using similar methods, we find that \( f''(0) = 0 \).
**Third Derivative:**
This will also yield \( f'''(0) = 0 \).
### Conclusion:
After analyzing all the functions, we find that:
- \( f(x) = |x^3| \) is **not** thrice differentiable at \( x = 0 \).
- \( f(x) = x^3 |x| \) is thrice differentiable at \( x = 0 \).
- \( f(x) = x |x| \sin^3(x) \) is thrice differentiable at \( x = 0 \).
- \( f(x) = x \tan^3(x) \) is thrice differentiable at \( x = 0 \).
Thus, the functions that are thrice differentiable at \( x = 0 \) are \( x^3 |x| \), \( x |x| \sin^3(x) \), and \( x \tan^3(x) \).
### Final Answer:
The functions that are thrice differentiable at \( x = 0 \) are:
- \( x^3 |x| \)
- \( x |x| \sin^3(x) \)
- \( x \tan^3(x) \)
