# Critical Points of Inflection

## Key Questions

• That is a good question! I had to revisit the definition in the Calculus book by Stewart, which states:

My answer to your question is no, a function does not need to be differentiable at a point of inflection; for example, the piecewise defined function

$f \left(x\right) = \left\{\begin{matrix}{x}^{2} & \mathmr{if} x < 0 \\ \sqrt{x} & \mathmr{if} x \ge 0\end{matrix}\right.$

is concave upward on $\left(- \infty , 0\right)$ and concave downward on $\left(0 , \infty\right)$ and is continuous at $x = 0$, so $\left(0 , 0\right)$ is an inflection point but not differentiable there.

• That is a good question! I had to revisit the definition in the Calculus book by Stewart, which states:

My answer to your question is yes, an inflection point could be an extremum; for example, the piecewise defined function

$f \left(x\right) = \left\{\begin{matrix}{x}^{2} & \mathmr{if} x < 0 \\ \sqrt{x} & \mathmr{if} x \ge 0\end{matrix}\right.$

is concave upward on $\left(- \infty , 0\right)$ and concave downward on $\left(0 , \infty\right)$ and is continuous at $x = 0$, so $\left(0 , 0\right)$ is an inflection point and a local (also global) minimum.