Let `f(x) ={{:( xe^(x), xle0),( x+x^(2)-x^(3), xgt0):}` then the correct statement is (a) f is continuous and differentiable for all ` x` (b) f is continuous but not differentiable at `x=0` (c) f is continuous and differentiable for all `x`. (d) f ' is continuous but not differentiable at `x=0`

Options

A. f is continuous and differentiable for all x,
B. f is continuous but not differentiable ata x=0
C. f is continuous and differentiable for all x.
D. f ' is continuous but not differentiable at x=0.

Answer

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

\[ f(x) = \begin{cases} x e^x & \text{for } x \leq 0 \\ x + x^2 - x^3 & \text{for } x > 0 \end{cases} \]

We will check the continuity and differentiability of \( f(x) \) at \( x = 0 \) and for all \( x \).

### Step 1: Check Continuity at \( x = 0 \)

To check if \( f(x) \) is continuous at \( x = 0 \), we need to evaluate the left-hand limit, right-hand limit, and the value of the function at that point.

1. **Left-hand limit** as \( x \to 0^- \):
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x e^x = 0 \cdot e^0 = 0 \]

2. **Right-hand limit** as \( x \to 0^+ \):
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x + x^2 - x^3) = 0 + 0 - 0 = 0 \]

3. **Value of the function** at \( x = 0 \):
\[ f(0) = 0 \cdot e^0 = 0 \]

Since the left-hand limit, right-hand limit, and the value of the function at \( x = 0 \) are all equal to 0, we conclude that \( f(x) \) is continuous at \( x = 0 \).

### Step 2: Check Differentiability at \( x = 0 \)

To check if \( f(x) \) is differentiable at \( x = 0 \), we need to find the left-hand derivative and right-hand derivative at that point.

1. **Left-hand derivative** at \( x = 0 \):
\[ f'(x) = \frac{d}{dx}(x e^x) = e^x + x e^x \]
Evaluating at \( x = 0 \):
\[ f'(0^-) = e^0 + 0 \cdot e^0 = 1 + 0 = 1 \]

2. **Right-hand derivative** at \( x = 0 \):
\[ f'(x) = \frac{d}{dx}(x + x^2 - x^3) = 1 + 2x - 3x^2 \]
Evaluating at \( x = 0 \):
\[ f'(0^+) = 1 + 2 \cdot 0 - 3 \cdot 0^2 = 1 \]

Since both the left-hand derivative and right-hand derivative at \( x = 0 \) are equal to 1, we conclude that \( f(x) \) is differentiable at \( x = 0 \).

### Step 3: Check Continuity and Differentiability for All \( x \)

Since \( f(x) \) is composed of two continuous functions (both parts are polynomials and exponential functions), and we have already established continuity and differentiability at \( x = 0 \), we conclude that \( f(x) \) is continuous and differentiable for all \( x \).

### Conclusion

Thus, the correct statement is:
- **(a)** \( f \) is continuous and differentiable for all \( x \).