#Study Notes: Local Extrema versus Global Extrema

#Table of Contents

1. Introduction
2. Local Extrema vs. Global Extrema
   • Definitions
   • Examples
3. Critical Points
   • Definition
   • Types of Critical Points
     • Local Minimums and Maximums
     • Points of Inflection
     • Non-Existent Derivative Points
   • Worked Examples
4. Practice Questions
5. Glossary
6. Summary and Key Takeaways
7. Exam Strategy

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#Introduction

In calculus, understanding the concepts of local and global extrema and critical points is essential for analyzing and interpreting the behavior of functions. This guide will help you distinguish between these concepts, identify critical points, and apply this knowledge effectively in exams.

#Local Extrema vs. Global Extrema

#Definitions

• Extrema: Refers to the maximum and minimum points on the graph of a function.
• Global (Absolute) Extrema: The highest (maximum) or lowest (minimum) value of the function over its entire domain.
• Local (Relative) Extrema: The highest or lowest value of the function within a specific interval of the domain.

Consider the function $f(x) = x^3 - 3x^2 + 4$: - The global minimum occurs at $x = -\infty$. - The local minima and maxima occur at points where the function's derivative equals zero.

#Examples

Here are some illustrative examples to clarify the concepts:

1. Global Maximum: The highest point on the entire graph.
2. Local Maximum: A peak within a specific interval but not necessarily the highest point overall.
3. Global Minimum: The lowest point on the entire graph.
4. Local Minimum: A trough within a specific interval but not necessarily the lowest point overall.

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#Critical Points

#Definition

A critical point is a point on the graph of a function where the first derivative is either zero or does not exist, provided the function itself is defined at that point.

#Types of Critical Points

#Local Minimums and Maximums

• At local minimums and maximums, the derivative is equal to zero: $$f'(x) = 0$$
• The tangent line at these points is horizontal.
• These points can also be global extrema.

For $y = (x - 2)^2$: - The local minimum at $(2, 0)$ is also the global minimum.

#Points of Inflection

• A point of inflection is where the graph changes concavity.
• Points of inflection are not local extrema.
• Only points of inflection where the first derivative is zero are critical points.

For $y = x^3$, the point $(0,0)$ is a point of inflection with the first derivative equal to zero.

#Non-Existent Derivative Points

• Critical points also occur where the derivative does not exist, provided the function is defined at that point.
• Consider $g(x) = x^{1/3}$: The function is defined at $x=0$, but the derivative does not exist there, making $x=0$ a critical point.

For $f(x) = (x - 2)^{1/3}$: - The function is defined at $x=2$, but the derivative does not exist, making $x=2$ a critical point.

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#Worked Examples

#Example 1: Polynomial Function

Problem: Find the coordinates of the critical points on the graph of the function $f(x) = 4x^3 - 30x^2 + 48x + 3$.

Solution:

1. Differentiate the function: $$f'(x) = 12x^2 - 60x + 48$$
2. Set the derivative equal to zero: $$12x^2 - 60x + 48 = 0$$ $$x^2 - 5x + 4 = 0$$ $$(x - 4)(x - 1) = 0$$
3. Solve for $x$: $$x = 4 \quad \text{or} \quad x = 1$$
4. Find the corresponding $y$ values: $$f(1) = 25$$ $$f(4) = -29$$

Critical Points: $(1, 25)$ and $(4, -29)$

#Example 2: Root Function

Problem: Find the coordinates of the critical points on the graph of the function $g(x) = (x - 3)^{2/3}$.

Solution:

1. Differentiate the function: $$g'(x) = \frac{2}{3}(x - 3)^{-1/3} = \frac{2}{3(x - 3)^{1/3}}$$
2. Identify points where the derivative is zero or undefined:
   • The derivative is undefined at $x = 3$.
3. Check if the function is defined at $x = 3$: $$g(3) = 0$$

Critical Point: $(3, 0)$

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#Practice Questions

#Multiple Choice

1. What is a critical point of a function?

   • A) A point where the second derivative is zero
   • B) A point where the first derivative is zero or does not exist
   • C) A point where the function is not defined

2. Which of the following is true about local extrema?

   • A) All local extrema are global extrema.
   • B) All global extrema are local extrema.
   • C) Local extrema occur only at the endpoints of the domain.

#Short-Answer

1. Find the critical points of the function $h(x) = x^4 - 4x^3 + 6x^2$.
2. Determine if the critical point $x = 0$ for the function $k(x) = x^{2/3}$ is a local extremum.

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#Glossary

• Extrema: Maximum and minimum points on the graph of a function.
• Global (Absolute) Extrema: The overall highest or lowest points of the function.
• Local (Relative) Extrema: The highest or lowest points within a specific interval.
• Critical Point: A point where the first derivative is zero or does not exist.
• Point of Inflection: A point where the graph changes concavity.

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#Summary and Key Takeaways

• Extrema: Understand the difference between local and global extrema.
• Critical Points: Know how to identify and classify critical points.
• Types of Critical Points: Distinguish between local minima, maxima, and points of inflection.
• Worked Examples: Practice finding critical points with polynomial and root functions.

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#Exam Strategy

• Identify Critical Points: Always start by finding where the first derivative is zero or undefined.
• Analyze Behavior: Use the second derivative test or concavity to determine the nature of critical points.
• Check Endpoints: For global extrema, always consider the endpoints of the domain if they exist.
• Practice: Regularly solve different types of problems to become proficient in identifying and analyzing critical points.

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By understanding and applying these concepts, you will be well-prepared to tackle questions involving local and global extrema and critical points in your exams. Keep practicing, and use these strategies to boost your confidence and performance.