Correct : -2.0

Explanation:
For a piecewise function to be differentiable everywhere, it must meet two conditions at the boundary point where the definition changes (at x = 1): it must be continuous, and its derivative must be continuous.

1. Condition 1: Continuity at x = 1
The Left-Hand Limit (LHL) must equal the Right-Hand Limit (RHL) at x = 1.
• LHL (as x approaches 1 from the left):
    LHL = lim_x→1^- (ax + b) = a(1) + b = a + b
• RHL (as x approaches 1 from the right):
    RHL = lim_x→1⁺ (x³ + x² + 1) = (1)³ + (1)² + 1 = 1 + 1 + 1 = 3

Equating the two for continuity gives us our first equation:
    a + b = 3  —  (Equation 1)

2. Condition 2: Differentiability at x = 1
The derivative of the function from the left must equal the derivative from the right at x = 1.
Let's find the derivative f'(x) for each piece:
• For x < 1: d/dx (ax + b) = a
• For x ≥ 1: d/dx (x³ + x² + 1) = 3x² + 2x

Now evaluate both side limits at x = 1:
• Left-Hand Derivative (LHD) = a
• Right-Hand Derivative (RHD) = 3(1)² + 2(1) = 3 + 2 = 5

Equating LHD and RHD for differentiability directly provides the value of a:
    a = 5

3. Solve for b:
Substitute the value of a into Equation 1:
    5 + b = 3
    b = 3 - 5 = -2

4. Conclusion:
The exact value of b is -2 (written as -2 or -2.0 rounded off to one decimal place).