Consider the given function \(f(x)\).
\[ f(x) = \begin{cases} ax + b & \text{for } x \lt 1 \\ x^{3} + x^{2} + 1 & \text{for } x \ge 1 \end{cases} \]
If the function is differentiable everywhere, the value of \(b\) must be ________. (rounded off to one decimal place)
Answer
-2.1 to -1.9
Explanation
A function that is differentiable at \(x = 1\) must be continuous there, and its left and right derivatives must be equal.
Continuity at \(x = 1\): \(a + b = 1 + 1 + 1 = 3\). Equal slopes: the left piece has slope \(a\), and the right piece has slope \(3x^{2} + 2x = 5\) at \(x = 1\), so \(a = 5\).
Hence \(b = 3 - 5 = -2\), that is, \(-2.0\).
Answer: −2.0