Continuity & Differentiability: Tricky Questions Solved (9)

Question

Find the values of $a$ and $b$ such that the function

$f(x) = \begin{cases} a x + 1 & \text{if } x \leq 1 \\ x^2 + b & \text{if } x > 1 \end{cases}$

is differentiable at $x = 1$.

Solution — Step by Step

Why This Works

Differentiability is a stronger condition than continuity. We need the function to be continuous (no jump) AND smooth (no kink) at the boundary. That gives two equations in two unknowns.

If we only impose differentiability via derivatives, we’d miss the continuity condition. Always check continuity FIRST when stitching piecewise functions.

Alternative Method

Compute the left and right derivatives from the limit definition:

$f'_-(1) = \lim_{h \to 0^-} \frac{f(1+h) - f(1)}{h}$

This is mechanically equivalent but slower. The piecewise-and-match approach above is cleaner.

Common Mistake

Students sometimes only impose differentiability ($a = 2$) and forget continuity. Without continuity, the “derivative” at the boundary doesn’t exist in the first place — derivatives require continuity as a prerequisite. So always do both.