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Chapter 3: Q. 6 (page 259)

Suppose fis a function that may be non-differentiable at some points. Can a point x=cbe 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.

Short Answer

Expert verified

For a function fwhich is non-differentiable, all three cases hold true.

Step by step solution

01

Step 1. Given Information.

The functionfis non-differentiable at some points.

02

Step 2. Local extremum and a critical point.

Checking for the case where the point cis local extremum and critical point.

Let the function be 8f(x)=x2which is a parabola and the graph is,

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

03

Step 3. An inflection point and a critical point.

Checking for the case where the point cis an inflection point and a critical point.

Let the function be f(x)=x3and the graph is,

The pointx=0is an inflection point and a critical point.

04

Step 4. An inflection point and a local extremum.

Checking for the case where the point cis a local extremum and an inflection point.

Considering the graph of a non-differentiable function,

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

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