Consider the function
`f(x)={{:(x^(2),"for "xgt2),(3x-2,"for "xle2):}`
which one of the following statements is correct in respect of the following function?

Options

A. f(x) is derivable but not continuous at x = 2
B. f(x) is continuous but not derivable at x = 2
C. f(x) is neither continuous nor derivable at x = 2
D. f(x) is continuous as well as derivable at x = 2

Answer

To analyze the function \( f(x) \) given by:

\[ f(x) = \begin{cases} x^2 & \text{for } x > 2 \\ 3x - 2 & \text{for } x \leq 2 \end{cases} \]

we need to check its continuity and differentiability at the point \( x = 2 \).

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

To check if the function is continuous at \( x = 2 \), we need to find the left-hand limit (LHL), right-hand limit (RHL), and the value of the function at that point.

1. **Left-hand limit (LHL)** as \( x \) approaches 2:
\[ \text{LHL} = \lim_{x \to 2^-} f(x) = f(2) = 3(2) - 2 = 6 - 2 = 4 \]

2. **Right-hand limit (RHL)** as \( x \) approaches 2:
\[ \text{RHL} = \lim_{x \to 2^+} f(x) = 2^2 = 4 \]

3. **Value of the function at \( x = 2 \)**:
\[ f(2) = 3(2) - 2 = 4 \]

Since LHL = RHL = \( f(2) = 4 \), the function is continuous at \( x = 2 \).

### Step 2: Check Differentiability at \( x = 2 \)

To check if the function is differentiable at \( x = 2 \), we need to find the left-hand derivative (LHD) and the right-hand derivative (RHD).

1. **Left-hand derivative (LHD)**:
The derivative of \( f(x) = 3x - 2 \) for \( x \leq 2 \) is:
\[ f'(x) = 3 \]
Thus,
\[ \text{LHD} = \lim_{h \to 0^-} \frac{f(2 + h) - f(2)}{h} = 3 \]

2. **Right-hand derivative (RHD)**:
The derivative of \( f(x) = x^2 \) for \( x > 2 \) is:
\[ f'(x) = 2x \]
Thus,
\[ \text{RHD} = \lim_{h \to 0^+} \frac{f(2 + h) - f(2)}{h} = \lim_{h \to 0^+} \frac{(2 + h)^2 - 4}{h} = \lim_{h \to 0^+} \frac{4 + 4h + h^2 - 4}{h} = \lim_{h \to 0^+} \frac{4h + h^2}{h} = 4 \]

Since LHD = 3 and RHD = 4, they are not equal. Therefore, the function is not differentiable at \( x = 2 \).

### Conclusion

The function \( f(x) \) is continuous at \( x = 2 \) but not differentiable at \( x = 2 \). Thus, the correct statement is:

**Option B**: \( f(x) \) is continuous but not differentiable at \( x = 2 \).

---