Consider the given function $f(x)$. $$f(x) = \begin{cases} ax + b & \text{for } x < 1 \\ x^3 + x^2 + 1 & \text{for } x \ge 1 \end{cases}$$ If the function is differentiable everywhere, the value of $b$ must be ________. (rounded off to one decimal place)

For the piecewise function $f(x)$ to be differentiable everywhere, it must be differentiable and continuous at the boundary point $x = 1$.

1. Differentiability at $x = 1$: The left-hand derivative (LHD) must equal the right-hand derivative (RHD) at $x = 1$:

Equating the derivatives gives:

2. Continuity at $x = 1$: Since differentiability implies continuity, the left-hand limit must equal the value of the function at $x = 1$:

Substitute $a = 5$ into the continuity equation:

Thus, the value of $b$ is -2.