A function is defined as follows
`f(x)={:{(x^(3),, x^(2)lt1),(x, , x^(2)ge 1):}` then function is

Options

A. Continuous at x=1
B. Differentiable at x=1
C. Continuous but not differentiable at x=1
D. None of these

Answer

To determine the continuity and differentiability of the function defined as:

\[ f(x) = \begin{cases} x^3 & \text{if } x^2 < 1 \\ x & \text{if } x^2 \geq 1 \end{cases} \]

we need to analyze the function at the point \( x = 1 \).

### Step 1: Determine the behavior of the function around \( x = 1 \)

1. **Identify the intervals**:
- For \( x^2 < 1 \) (which means \( -1 < x < 1 \)), we have \( f(x) = x^3 \).
- For \( x^2 \geq 1 \) (which means \( x \leq -1 \) or \( x \geq 1 \)), we have \( f(x) = x \).

### Step 2: Check continuity at \( x = 1 \)

To check continuity at \( x = 1 \), we need to evaluate the left-hand limit, the right-hand limit, and the function value at that point.

2. **Left-hand limit**:
\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x^3 = 1^3 = 1 \]

3. **Right-hand limit**:
\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} x = 1 \]

4. **Function value at \( x = 1 \)**:
\[ f(1) = 1 \]

5. **Conclusion on continuity**:
Since the left-hand limit, right-hand limit, and the function value at \( x = 1 \) are all equal:
\[ \lim_{x \to 1^-} f(x) = f(1) = \lim_{x \to 1^+} f(x) = 1 \]
Therefore, \( f(x) \) is continuous at \( x = 1 \).

### Step 3: Check differentiability at \( x = 1 \)

To check differentiability at \( x = 1 \), we need to evaluate the left-hand derivative and the right-hand derivative.

6. **Left-hand derivative**:
\[ f'(x) = \frac{d}{dx}(x^3) = 3x^2 \]
Thus,
\[ \lim_{x \to 1^-} f'(x) = \lim_{x \to 1^-} 3x^2 = 3(1^2) = 3 \]

7. **Right-hand derivative**:
\[ f'(x) = \frac{d}{dx}(x) = 1 \]
Thus,
\[ \lim_{x \to 1^+} f'(x) = 1 \]

8. **Conclusion on differentiability**:
Since the left-hand derivative and right-hand derivative are not equal:
\[ \lim_{x \to 1^-} f'(x) = 3 \quad \text{and} \quad \lim_{x \to 1^+} f'(x) = 1 \]
Therefore, \( f(x) \) is not differentiable at \( x = 1 \).

### Final Conclusion

The function \( f(x) \) is continuous at \( x = 1 \) but not differentiable at \( x = 1 \).

### Summary of the Answer
The correct option is: **Continuous at \( x = 1 \) but not differentiable at \( x = 1 \)**.

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