Determine if the piecewise defined function is differentiable at the origin.
-$$f ( x ) = \left\{ \begin{array} { l l }
4 x - 6 & \text { if } x

Question

Determine if the piecewise defined function is differentiable at the origin.
-$$f ( x ) = \left\{ \begin{array} { l l } 
4 x - 6 & \text { if } x < 0 \
x ^ { 2 } + 6 x + 6 & \text { if } x \geq 0
\end{array} \right.$$
A) Not differentiable
B) Differentiable

Quizplus · Accepted Answer

For a function to be differentiable at a point, it must be continuous at that point and its left-hand derivative must equal its right-hand derivative at that point. First, check for continuity at $x = 0$. For $x < 0$, $f(x) = 4x - 6$, so $f(0^-) = -6$. For $x \geq 0$, $f(x) = x^2 + 6x + 6$, so $f(0^+) = 6$. Since $f(0^-) \neq f(0^+)$, the function is not continuous at $x = 0$, and thus not differentiable at $x = 0$.

Determine If the Piecewise Defined Function Is Differentiable at the Origin