#Critical Points of Implicit Relations

#Table of Contents

Introduction to Critical Points
Identifying Critical Points in Implicit Equations
Horizontal and Vertical Tangents
Worked Example
Practice Questions
Glossary
Summary and Key Takeaways

#Introduction to Critical Points

Critical points are crucial in determining the behavior of functions. They are points on the graph of a function where the derivative is zero or undefined. These points help in identifying local maxima, local minima, and points of inflection.

#Identifying Critical Points in Implicit Equations

Implicit equations can have critical points, which are defined similarly to those in explicit functions.

#Key Concepts

Key Concept

Critical Point: A point where the derivative of the function is either zero or does not exist.

The application of first and second derivatives to classify the nature of points on a graph can be extended to implicit functions.

#Classifying Critical Points

Here is a summary of how the first and second derivatives are used to classify critical points:

Type of Point	First Derivative	Second Derivative
Local Minimum	Zero	Positive or zero
Local Maximum	Zero	Negative or zero
Point of Inflection (Critical)	Zero	Zero
Point of Inflection (Non-critical)	Non-zero	Zero

Common Mistake

Second derivative equal to zero is not enough for a point to be a point of inflection. The second derivative must change sign at the point as well.

#Horizontal and Vertical Tangents...

Previous Topic - Second Derivatives of Implicit Functions Next Topic - Indefinite Integrals

Behaviors of Implicit Relations

David Brown