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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1:46 minutes

Problem 4.3.5a

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) = (x − 1)(x + 2)(x − 3)

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. Since f'(x) = (x - 1)(x + 2)(x - 3), we focus on where this expression equals zero.

Set f'(x) = 0: (x - 1)(x + 2)(x - 3) = 0. This equation is satisfied when any of the factors is zero.

Solve each factor for x: x - 1 = 0, x + 2 = 0, and x - 3 = 0. This gives us the potential critical points.

The solutions to these equations are x = 1, x = -2, and x = 3. These are the critical points of the function f.

Verify that these points are indeed critical points by ensuring that f'(x) changes sign around these values, indicating a change in the behavior of the function f.

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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 local maxima, minima, or points of inflection. To find critical points, set the derivative equal to zero and solve for the variable, which in this case involves solving f′(x) = (x − 1)(x + 2)(x − 3) = 0.

Factoring Polynomials

Factoring polynomials is a method used to simplify expressions and solve equations. It involves expressing a polynomial as a product of its factors. For the derivative f′(x) = (x − 1)(x + 2)(x − 3), the factors are already given, making it straightforward to find the roots by setting each factor equal to zero: x = 1, x = -2, and x = 3.

Roots of Equations

The roots of an equation are the values of the variable that satisfy the equation, making it equal to zero. In the context of derivatives, finding the roots of f′(x) helps identify the critical points of the function f(x). For f′(x) = (x − 1)(x + 2)(x − 3), the roots are x = 1, x = -2, and x = 3, which are the critical points of the function.

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Analyzing Functions from Derivatives