The function `f(x)={ 1-2x+3x^2-4x^3; x!=-1, 1; x=-1}` discuss the continuity and differentiablilty at `x=-1`.

Options

A. continuous and derivable at `x=-1`
B. Neither continuous nor derivable at `x=-1`
C. Continuous but not derivasble at `x=-1`
D. continuous but not derivable at `x=0`

Explanation

To determine the continuity and differentiability of the function \( f(x) \) at \( x = -1 \), we will follow these steps:

### Step 1: Define the function
The function is defined as:
\[ f(x) = \begin{cases} 1 - 2x + 3x^2 - 4x^3 & \text{if } x \neq -1 \\ 1 & \text{if } x = -1 \end{cases} \]

### Step 2: Check continuity at \( x = -1 \)
To check for continuity at \( x = -1 \), we need to evaluate:
1. \( f(-1) \)
2. \( \lim_{x \to -1} f(x) \)
3. Compare \( f(-1) \) with \( \lim_{x \to -1} f(x) \)

**Calculate \( f(-1) \)**:
\[ f(-1) = 1 \]

**Calculate \( \lim_{x \to -1} f(x) \)**:
For \( x \neq -1 \):
\[ f(x) = 1 - 2x + 3x^2 - 4x^3 \]
We need to find:
\[ \lim_{x \to -1} (1 - 2x + 3x^2 - 4x^3) \]
Substituting \( x = -1 \):
\[ = 1 - 2(-1) + 3(-1)^2 - 4(-1)^3 = 1 + 2 + 3 + 4 = 10 \]

**Compare \( f(-1) \) and \( \lim_{x \to -1} f(x) \)**:
\[ f(-1) = 1 \quad \text{and} \quad \lim_{x \to -1} f(x) = 10 \]
Since \( f(-1) \neq \lim_{x \to -1} f(x) \), \( f(x) \) is not continuous at \( x = -1 \).

### Step 3: Check differentiability at \( x = -1 \)
For differentiability, we need to check if the derivative exists at that point. The derivative at a point is given by:
\[ f'(-1) = \lim_{h \to 0} \frac{f(-1 + h) - f(-1)}{h} \]
Substituting \( f(-1) = 1 \):
\[ f'(-1) = \lim_{h \to 0} \frac{f(-1 + h) - 1}{h} \]
For \( h \) close to 0 (but not equal to -1):
\[ f(-1 + h) = 1 - 2(-1 + h) + 3(-1 + h)^2 - 4(-1 + h)^3 \]
Calculating \( f(-1 + h) \):
\[ = 1 + 2 - 2h + 3(1 - 2h + h^2) - 4(-1 + h)^3 \]
This calculation can be complex, but we can simplify it by evaluating the limit:
\[ = \lim_{h \to 0} \frac{(10 - 1)}{h} = \lim_{h \to 0} \frac{9}{h} \]
As \( h \to 0 \), this limit approaches infinity. Therefore, \( f'(-1) \) does not exist.

### Conclusion
Since \( f(x) \) is neither continuous nor differentiable at \( x = -1 \), we conclude:
- \( f(x) \) is not continuous at \( x = -1 \).
- \( f(x) \) is not differentiable at \( x = -1 \).