Q. 6 - Calculus

Question

Suppose $f$ is a function that may be non-differentiable at some points. Can a point $x=c$ be both a local extremum and a critical point of such a function $f$ ? Both an inflection point and a critical point? Both an inflection point and a local extremum? Sketch examples, or explain why such a point cannot exist.

Step-by-Step Solution

Answer

For a function $f$ which is non-differentiable, all three cases hold true.

Step 1. Given Information.

The function $f$ is non-differentiable at some points.

Step 2. Local extremum and a critical point.

Checking for the case where the  point $c$ is local extremum and critical point.

Let the function be $8f\left(x\right)=x^2$ which is a parabola and the graph is,

The point $x=0$ is a local extremum cum critical point.

Step 3. An inflection point and a critical point.

Checking for the case where the point $c$ is an inflection point and a critical point.

Let the function be $f\left(x\right)=x^3$ and the graph is,

The point $x=0$ is an inflection point and a critical point.

Step 4. An inflection point and a local extremum.

Checking for the case where the point $c$ is a local extremum and an inflection point.

Considering the graph of a non-differentiable function,

The point $x=0$ is a local extremum and an inflection point.