If the function and the derivative of the function `f(x)` is everywhere continuous and is given by
`f(x)={:{(bx^(2)+ax+4", for " x ge -1),(ax^(2)+b", for " x lt -1):}`, then

Options

A. `a=2,b=3`
B. `a=1,b=1`
C. `a=-2,b=-3`
D. `a=-3,b=-2`

Answer

To solve the problem, we need to ensure that the function \( f(x) \) is continuous and differentiable at \( x = -1 \). The function is defined piecewise as follows:

\[ f(x) = \begin{cases} bx^2 + ax + 4 & \text{for } x \ge -1 \\ ax^2 + b & \text{for } x < -1 \end{cases} \]

### Step 1: Ensure Continuity at \( x = -1 \)

For \( f(x) \) to be continuous at \( x = -1 \), the left-hand limit (as \( x \) approaches -1 from the left) must equal the right-hand limit (as \( x \) approaches -1 from the right):

\[ f(-1^-) = f(-1^+) \]

Calculating \( f(-1^-) \):

\[ f(-1^-) = a(-1)^2 + b = a + b \]

Calculating \( f(-1^+) \):

\[ f(-1^+) = b(-1)^2 + a(-1) + 4 = b - a + 4 \]

Setting these equal gives us the equation:

\[ a + b = b - a + 4 \]

### Step 2: Simplify the Continuity Equation

Rearranging the equation:

\[ a + b = b - a + 4 \implies a + a = 4 \implies 2a = 4 \implies a = 2 \]

### Step 3: Ensure Differentiability at \( x = -1 \)

For \( f(x) \) to be differentiable at \( x = -1 \), the left-hand derivative must equal the right-hand derivative:

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

Calculating \( f'(-1^-) \):

\[ f'(-1^-) = \frac{d}{dx}(ax^2 + b) = 2ax \quad \text{(for } x < -1\text{)} \]
Substituting \( x = -1 \):

\[ f'(-1^-) = 2a(-1) = -2a \]

Calculating \( f'(-1^+) \):

\[ f'(-1^+) = \frac{d}{dx}(bx^2 + ax + 4) = 2bx + a \quad \text{(for } x \ge -1\text{)} \]
Substituting \( x = -1 \):

\[ f'(-1^+) = 2b(-1) + a = -2b + a \]

Setting these equal gives us the equation:

\[ -2a = -2b + a \]

### Step 4: Simplify the Differentiability Equation

Rearranging the equation:

\[ -2a - a = -2b \implies -3a = -2b \implies 3a = 2b \implies b = \frac{3}{2}a \]

### Step 5: Substitute the Value of \( a \)

Now substituting \( a = 2 \):

\[ b = \frac{3}{2}(2) = 3 \]

### Conclusion

Thus, the values of \( a \) and \( b \) are:

\[ a = 2, \quad b = 3 \]

### Final Answer

The values of \( a \) and \( b \) are \( a = 2 \) and \( b = 3 \).

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