Suppose $f$ is a piecewise-defined function, equal to $g ( x ) \text { if } x

Question

Suppose $f$ is a piecewise-defined function, equal to $g ( x ) \text { if } x < 3 \text {, and } h ( x ) \text { if } x \geq 3 \text {, }$ where $g \text { and } h$ are continuous and differentiable everywhere. If $g ^ { \prime } ( 3 ) = h ^ { \prime } ( 3 )$ is the function $f$ differentiable at $x = 3 ?$ Explain why or why not.

Quizplus · Accepted Answer

For a piecewise function to be differentiable at a point where the definition changes (in this case, $x = 3$), it must be both continuous and have matching derivatives from both sides at that point. Since $g$ and $h$ are continuous and differentiable everywhere, and their derivatives are equal at $x = 3$, $f$ is continuous and has a matching derivative from both sides at $x = 3$. Therefore, $f$ is differentiable at $x = 3$.

Suppose fff Is a Piecewise-Defined Function, Equal To g(x) if x<3, and h(x) if x≥3, g ( x ) \text { if } x < 3 \text {, and } h ( x ) \text { if } x \geq 3 \text {, }g(x) if x<3, and h(x) if x≥3,

Suppose $f$ Is a Piecewise-Defined Function, Equal To $g ( x ) \text { if } x < 3 \text {, and } h ( x ) \text { if } x \geq 3 \text {, }$