Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

The First Derivative Test

Calculus

5. Graphical Applications of Derivatives

The First Derivative Test

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2:57 minutes

Problem 4.3.9a

Textbook Question

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

a. What are the critical points of f?

f′(x) = 1− 4/x², x ≠ 0

Verified step by step guidance

To find the critical points of the function f, we need to determine where the derivative f'(x) is equal to zero or undefined. Critical points occur where the derivative changes sign or is undefined.

Given the derivative f'(x) = 1 - 4/x², we first set f'(x) equal to zero to find where the derivative changes sign: 1 - 4/x² = 0.

Solve the equation 1 - 4/x² = 0 for x. This involves isolating x² by adding 4/x² to both sides, resulting in 1 = 4/x².

Next, solve for x² by multiplying both sides by x², giving x² = 4. Then, take the square root of both sides to find the values of x: x = ±2.

Since x ≠ 0, the critical points are x = 2 and x = -2. These are the points where the derivative is zero, indicating potential maxima, minima, or points of inflection.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are significant because they can indicate potential local maxima, minima, or points of inflection. To find critical points, set the derivative equal to zero and solve for x, or identify where the derivative does not exist.

Derivative Analysis

Analyzing the derivative of a function helps determine the behavior of the original function. The sign of the derivative indicates whether the function is increasing or decreasing. In this context, f′(x) = 1 - 4/x² must be analyzed to find where it equals zero or is undefined, revealing critical points.

Rational Functions

A rational function is a ratio of two polynomials. The derivative given, f′(x) = 1 - 4/x², is a rational function. Understanding how to manipulate and solve rational functions is crucial for finding where the derivative equals zero or is undefined, which helps identify critical points.

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Analyzing Functions from Derivatives