Let `f (x)=[{:((3x-x ^(2))/(2),,, x lt 2),([x-1],,, 2 le x lt 3),( x ^(2) -8x+17,,, x ge3):}:` then which of the following hold(s) good ?

Options

A. `lim _(xto2) f(x) =1`
B. `f (x)` is differentiable at `x=2`
C. `f (x)` is continous at `x=2`
D. `f (x)` is discontinous at `x =3`

Answer

To solve the problem step by step, we need to analyze the piecewise function \( f(x) \) given by:

\[ f(x) = \begin{cases} \frac{3x - x^2}{2} & \text{if } x < 2 \\ x - 1 & \text{if } 2 \leq x < 3 \\ x^2 - 8x + 17 & \text{if } x \geq 3 \end{cases} \]

We will check the continuity and differentiability of \( f(x) \) at the points \( x = 2 \) and \( x = 3 \).

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

To check continuity at \( x = 2 \), we need to find:

1. \( \lim_{x \to 2^-} f(x) \)
2. \( \lim_{x \to 2^+} f(x) \)
3. \( f(2) \)

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

**Right-hand limit**:
\[ \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x - 1) = 2 - 1 = 1 \]

**Value at \( x = 2 \)**:
\[ f(2) = 2 - 1 = 1 \]

Since \( \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2) = 1 \), the function is continuous at \( x = 2 \).

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

To check differentiability at \( x = 2 \), we need to find the left-hand and right-hand derivatives.

**Left-hand derivative**:
\[ f'(x) = \frac{d}{dx}\left(\frac{3x - x^2}{2}\right) = \frac{3 - 2x}{2} \]
\[ \lim_{x \to 2^-} f'(x) = \frac{3 - 2(2)}{2} = \frac{3 - 4}{2} = \frac{-1}{2} \]

**Right-hand derivative**:
\[ f'(x) = \frac{d}{dx}(x - 1) = 1 \]
\[ \lim_{x \to 2^+} f'(x) = 1 \]

Since the left-hand derivative \( \frac{-1}{2} \) is not equal to the right-hand derivative \( 1 \), \( f(x) \) is not differentiable at \( x = 2 \).

### Step 3: Check continuity at \( x = 3 \)

To check continuity at \( x = 3 \), we need to find:

1. \( \lim_{x \to 3^-} f(x) \)
2. \( \lim_{x \to 3^+} f(x) \)
3. \( f(3) \)

**Left-hand limit**:
\[ \lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} (x - 1) = 3 - 1 = 2 \]

**Right-hand limit**:
\[ \lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} (x^2 - 8x + 17) = (3)^2 - 8(3) + 17 = 9 - 24 + 17 = 2 \]

**Value at \( x = 3 \)**:
\[ f(3) = 3^2 - 8(3) + 17 = 9 - 24 + 17 = 2 \]

Since \( \lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3) = 2 \), the function is continuous at \( x = 3 \).

### Step 4: Check differentiability at \( x = 3 \)

**Left-hand derivative**:
\[ \lim_{x \to 3^-} f'(x) = 1 \]

**Right-hand derivative**:
\[ f'(x) = \frac{d}{dx}(x^2 - 8x + 17) = 2x - 8 \]
\[ \lim_{x \to 3^+} f'(x) = 2(3) - 8 = 6 - 8 = -2 \]

Since the left-hand derivative \( 1 \) is not equal to the right-hand derivative \( -2 \), \( f(x) \) is not differentiable at \( x = 3 \).

### Conclusion

- \( f(x) \) is continuous at \( x = 2 \) and \( x = 3 \).
- \( f(x) \) is not differentiable at \( x = 2 \) and \( x = 3 \).
- The function is discontinuous at \( x = 3 \).

### Final Answer

The correct options are:
1. \( \lim_{x \to 2} f(x) = 1 \) (True)
2. \( f(x) \) is differentiable at \( x = 2 \) (False)
3. \( f(x) \) is continuous at \( x = 2 \) (True)
4. \( f(x) \) is discontinuous at \( x = 3 \) (True)