Given the continuous fuunction
` y= f(x) ={{:( x^(2) +10x+8, x le -2),( ax^(2)+bx +c, -2ltxlt 0","a ne 0) ,( x^(2) + 2x, xge 0):}`
if a line L touches the graph of y=f(x) at three points , then
if y= f(x) is differentiable at x=0, then the value of B

Options

A. is -1
B. is 2
C. is 4
D. connot be determined

Answer

To solve the problem step by step, we need to ensure that the function \( f(x) \) is continuous and differentiable at \( x = 0 \). We will analyze the function piecewise and find the necessary conditions.

### Step 1: Define the function piecewise
The function \( f(x) \) is defined as follows:
- For \( x \leq -2 \): \( f(x) = x^2 + 10x + 8 \)
- For \( -2 < x < 0 \): \( f(x) = ax^2 + bx + c \)
- For \( x \geq 0 \): \( f(x) = x^2 + 2x \)

### Step 2: Ensure continuity at \( x = -2 \)
To ensure continuity at \( x = -2 \), we need:
\[ \lim_{x \to -2^-} f(x) = \lim_{x \to -2^+} f(x) \]
Calculating the left-hand limit:
\[ f(-2) = (-2)^2 + 10(-2) + 8 = 4 - 20 + 8 = -8 \]
Now for the right-hand limit:
\[ f(-2) = a(-2)^2 + b(-2) + c = 4a - 2b + c \]
Setting these equal for continuity:
\[ 4a - 2b + c = -8 \quad \text{(1)} \]

### Step 3: Ensure continuity at \( x = 0 \)
Next, we ensure continuity at \( x = 0 \):
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) \]
Calculating the left-hand limit:
\[ f(0) = a(0)^2 + b(0) + c = c \]
Now for the right-hand limit:
\[ f(0) = (0)^2 + 2(0) = 0 \]
Setting these equal for continuity:
\[ c = 0 \quad \text{(2)} \]

### Step 4: Substitute \( c \) back into equation (1)
Substituting \( c = 0 \) into equation (1):
\[ 4a - 2b + 0 = -8 \]
This simplifies to:
\[ 4a - 2b = -8 \quad \text{(3)} \]

### Step 5: Ensure differentiability at \( x = 0 \)
For differentiability at \( x = 0 \), the left-hand derivative must equal the right-hand derivative:
\[ \lim_{x \to 0^-} f'(x) = \lim_{x \to 0^+} f'(x) \]
Calculating the left-hand derivative:
\[ f'(x) = 2ax + b \quad \Rightarrow \quad \lim_{x \to 0^-} f'(x) = b \]
Calculating the right-hand derivative:
\[ f'(x) = 2x + 2 \quad \Rightarrow \quad \lim_{x \to 0^+} f'(x) = 2 \]
Setting these equal for differentiability:
\[ b = 2 \quad \text{(4)} \]

### Step 6: Solve for \( a \) using equation (3)
Substituting \( b = 2 \) into equation (3):
\[ 4a - 2(2) = -8 \]
This simplifies to:
\[ 4a - 4 = -8 \]
Adding 4 to both sides:
\[ 4a = -4 \]
Dividing by 4:
\[ a = -1 \]

### Final Result
The value of \( b \) is:
\[ \boxed{2} \]