If `f(x) = {{:(A+Bx^(2)",",x lt 1),(3Ax - B+2",",x ge 1):}`, then A and B, so that f(x) is differentiabl at x = 1, are

Options

A. `-2, 3`
B. `2, -3`
C. 2, 3
D. `-2, -3`

Answer

To determine the values of \( A \) and \( B \) such that the function \( f(x) \) is differentiable at \( x = 1 \), we will follow these steps:

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

For \( f(x) \) to be differentiable at \( x = 1 \), it must first be continuous at that point. This means that the left-hand limit as \( x \) approaches 1 must equal the right-hand limit at \( x = 1 \).

The function is defined as:
- \( f(x) = A + Bx^2 \) for \( x < 1 \)
- \( f(x) = 3Ax - B + 2 \) for \( x \geq 1 \)

Calculating the left-hand limit:
\[ \lim_{x \to 1^-} f(x) = A + B(1^2) = A + B \]

Calculating the right-hand limit:
\[ \lim_{x \to 1^+} f(x) = 3A(1) - B + 2 = 3A - B + 2 \]

Setting these two limits equal for continuity:
\[ A + B = 3A - B + 2 \]

### Step 2: Solve for \( A \) and \( B \)

Rearranging the equation:
\[ A + B = 3A - B + 2 \]
\[ A + B + B = 3A + 2 \]
\[ A + 2B = 3A + 2 \]
\[ 2B = 3A - A + 2 \]
\[ 2B = 2A + 2 \]
Dividing by 2:
\[ B = A + 1 \quad \text{(Equation 1)} \]

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

Next, we need to ensure that the derivatives from both sides are equal at \( x = 1 \).

Calculating the derivative for \( x < 1 \):
\[ f'(x) = \frac{d}{dx}(A + Bx^2) = 2Bx \]
Thus,
\[ \lim_{x \to 1^-} f'(x) = 2B(1) = 2B \]

Calculating the derivative for \( x \geq 1 \):
\[ f'(x) = \frac{d}{dx}(3Ax - B + 2) = 3A \]
Thus,
\[ \lim_{x \to 1^+} f'(x) = 3A \]

Setting these two derivatives equal for differentiability:
\[ 2B = 3A \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2

Substituting \( B = A + 1 \) into Equation 2:
\[ 2(A + 1) = 3A \]
Expanding:
\[ 2A + 2 = 3A \]
Rearranging:
\[ 2 = 3A - 2A \]
\[ A = 2 \]

### Step 5: Find \( B \)

Now substituting \( A = 2 \) back into Equation 1:
\[ B = 2 + 1 = 3 \]

### Final Answer

Thus, the values of \( A \) and \( B \) that make \( f(x) \) differentiable at \( x = 1 \) are:
\[ A = 2, \quad B = 3 \]