Let `f(x)=a+b|x|+c|x|^4` , where `a ,\ b ,` and `c` are real constants. Then, `f(x)` is differentiable at `x=0` , if

Options

A. `a=0`
B. `b=0`
C. `c=0`
D. none of these

Answer

To determine the conditions under which the function \( f(x) = a + b|x| + c|x|^4 \) is differentiable at \( x = 0 \), we need to follow a systematic approach involving continuity and the definition of differentiability.

### Step-by-Step Solution:

1. **Check Continuity at \( x = 0 \)**:
For \( f(x) \) to be differentiable at \( x = 0 \), it must first be continuous at that point. This means:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0) \]

2. **Evaluate \( f(0) \)**:
\[ f(0) = a + b|0| + c|0|^4 = a \]

3. **Evaluate the Left-Hand Limit \( \lim_{x \to 0^-} f(x) \)**:
For \( x < 0 \), \( |x| = -x \):
\[ f(x) = a + b(-x) + c(-x)^4 = a - bx + cx^4 \]
Thus,
\[ \lim_{x \to 0^-} f(x) = a + 0 + 0 = a \]

4. **Evaluate the Right-Hand Limit \( \lim_{x \to 0^+} f(x) \)**:
For \( x > 0 \), \( |x| = x \):
\[ f(x) = a + bx + cx^4 \]
Thus,
\[ \lim_{x \to 0^+} f(x) = a + 0 + 0 = a \]

5. **Conclusion on Continuity**:
Since both limits equal \( f(0) \):
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0) = a \]
Therefore, \( f(x) \) is continuous at \( x = 0 \).

6. **Check Differentiability at \( x = 0 \)**:
For \( f(x) \) to be differentiable at \( x = 0 \), we need:
\[ \lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{f(-h) - f(0)}{-h} \]

7. **Calculate Left-Hand Derivative**:
\[ \lim_{h \to 0} \frac{f(-h) - a}{-h} = \lim_{h \to 0} \frac{(a + b(-h) + c(-h)^4) - a}{-h} = \lim_{h \to 0} \frac{-bh + ch^4}{-h} = \lim_{h \to 0} (b - ch^3) = b \]

8. **Calculate Right-Hand Derivative**:
\[ \lim_{h \to 0} \frac{f(h) - a}{h} = \lim_{h \to 0} \frac{(a + bh + ch^4) - a}{h} = \lim_{h \to 0} \frac{bh + ch^4}{h} = \lim_{h \to 0} (b + ch^3) = b \]

9. **Set Left-Hand Derivative Equal to Right-Hand Derivative**:
For differentiability:
\[ b = b \]
This is always true, but we need to check the behavior of the terms involving \( c \).

10. **Final Condition**:
The limit involving \( c \) must also hold, which leads to:
\[ -b = b \Rightarrow 2b = 0 \Rightarrow b = 0 \]

### Conclusion:
Thus, the function \( f(x) \) is differentiable at \( x = 0 \) if \( b = 0 \).