The function f (x) is defined as under :
`f(x)={{:(3^(x), -1 le x le 1),(4-x, 1 lt x lt 4):}`
The above 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 properties of the function \( f(x) \) defined as:

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

we need to check its continuity and differentiability at the point \( x = 1 \).

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

To check continuity at \( x = 1 \), we need to verify that:

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

**Left-hand limit:**

\[ \lim_{x \to 1^-} f(x) = \lim_{h \to 0} f(1 - h) = \lim_{h \to 0} 3^{1 - h} = 3^1 = 3 \]

**Right-hand limit:**

\[ \lim_{x \to 1^+} f(x) = \lim_{h \to 0} f(1 + h) = \lim_{h \to 0} (4 - (1 + h)) = \lim_{h \to 0} (3 - h) = 3 \]

**Function value at \( x = 1 \):**

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

Since both the left-hand limit and right-hand limit equal the function value at \( x = 1 \):

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

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

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

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

**Left-hand derivative:**

\[ f'(1^-) = \lim_{h \to 0} \frac{f(1 - h) - f(1)}{-h} = \lim_{h \to 0} \frac{3^{1 - h} - 3}{-h} \]

Using the formula for the derivative of an exponential function, we can rewrite this as:

\[ = \lim_{h \to 0} \frac{3^{1 - h} - 3}{-h} = -3 \lim_{h \to 0} \frac{3^{-h} - 1}{h} = -3 \cdot \ln(3) \]

**Right-hand derivative:**

\[ f'(1^+) = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h} = \lim_{h \to 0} \frac{(4 - (1 + h)) - 3}{h} = \lim_{h \to 0} \frac{3 - h - 3}{h} = \lim_{h \to 0} \frac{-h}{h} = -1 \]

### Conclusion

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

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

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

### Final Answer

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

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