Let `f(x)={{:(3^x","-1lexle1),(4-x","1ltxle4):}` then

Options

A. `f(x)` is continuous and derivable at `x=1`
B. `f(x)` is discontinuous and non-derivable at `x=1`
C. `f(x)` is continuous and non-derivable at `x=1` because `f'(1^(+)) = -1` and `f'(1^(-))=3 l n3`
D. `f(x)` is continuous and non-derivable at `x=1` because `f'(1^(+))=-1` and `f'(1^(-))=1`

Explanation

To solve the problem, we need to analyze the function \( f(x) \) defined piecewise as follows:

\[ f(x) = \begin{cases} 3^x & \text{for } -1 \leq x \leq 1 \\ 4 - x & \text{for } 1 < x \leq 4 \end{cases} \]

We need to check the continuity and differentiability of \( f(x) \) at \( x = 1 \).

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

For \( f(x) \) to be continuous at \( x = 1 \), the following condition must hold:

\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) \]

#### Calculate \( f(1) \)

Using the first piece of the function:

\[ f(1) = 3^1 = 3 \]

#### Calculate \( \lim_{x \to 1^-} f(x) \)

As \( x \) approaches 1 from the left, we use the first piece of the function:

\[ \lim_{x \to 1^-} f(x) = 3^1 = 3 \]

#### Calculate \( \lim_{x \to 1^+} f(x) \)

As \( x \) approaches 1 from the right, we use the second piece of the function:

\[ \lim_{x \to 1^+} f(x) = 4 - 1 = 3 \]

Since both limits and \( f(1) \) are equal:

\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) = 3 \]

Thus, \( f(x) \) is continuous at \( x = 1 \).

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

For \( f(x) \) to be differentiable at \( x = 1 \), the left-hand derivative and right-hand derivative must be equal.

#### Calculate Left-Hand Derivative \( f'(1^-) \)

Using the first piece of the function:

\[ f'(x) = \frac{d}{dx}(3^x) = \ln(3) \cdot 3^x \]

At \( x = 1 \):

\[ f'(1^-) = \ln(3) \cdot 3^1 = 3 \ln(3) \]

#### Calculate Right-Hand Derivative \( f'(1^+) \)

Using the second piece of the function:

\[ f'(x) = \frac{d}{dx}(4 - x) = -1 \]

At \( x = 1 \):

\[ f'(1^+) = -1 \]

### Step 3: Compare the Derivatives

Now we compare the left-hand and right-hand derivatives:

\[ f'(1^-) = 3 \ln(3) \quad \text{and} \quad f'(1^+) = -1 \]

Since \( 3 \ln(3) \neq -1 \), the left-hand derivative does not equal the right-hand derivative.

Thus, \( f(x) \) is not differentiable at \( x = 1 \).

### Conclusion

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

### Final Answer

The correct option is that \( f(x) \) is continuous but not differentiable at \( x = 1 \).