Determine if the piecewise defined function is differentiable at the origin.
-$$g ( x ) = \left\{ \begin{array} { l l }
x ^ { 7 / 8 } & \text { if } x \geq 0 \
x ^ { 1 / 8 } & \text { if } x

Question

Determine if the piecewise defined function is differentiable at the origin.
-$$g ( x ) = \left\{ \begin{array} { l l } 
x ^ { 7 / 8 } & \text { if } x \geq 0 \
x ^ { 1 / 8 } & \text { if } x < 0
\end{array} \right.$$
A) Not d ifferentiable
B) Differentiable

Quizplus · Accepted Answer

For a function to be differentiable at a point, it must be continuous at that point and its derivative must also be continuous at that point. The given function is continuous at the origin since both $x^{7/8}$ and $x^{1/8}$ approach 0 as $x$ approaches 0 from their respective sides. However, when we take the derivative of each piece, we get $\frac{7}{8}x^{-1/8}$ for $x \geq 0$ and $\frac{1}{8}x^{-7/8}$ for $x < 0$. Evaluating these derivatives at $x = 0$ is problematic because they both involve division by zero, indicating that the derivative does not exist at the origin. Therefore, the function is not differentiable at the origin.

Determine If the Piecewise Defined Function Is Differentiable at the Origin