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

Intro to Extrema

Calculus

5. Graphical Applications of Derivatives

Intro to Extrema

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2:25 minutes

Problem 4.1.13

Textbook Question

Finding Extrema from Graphs

In Exercises 11–14, match the table with a graph.

Verified step by step guidance

Step 1: Analyze the table provided. It indicates the behavior of the derivative f'(x) at specific points: at x = a, f'(x) does not exist; at x = b, f'(x) = 0; and at x = c, f'(x) = -2.

Step 2: Understand the implications of the derivative values: - 'f'(x) does not exist' at x = a suggests a sharp corner or cusp in the graph at x = a. - 'f'(x) = 0' at x = b indicates a horizontal tangent line, which is typically a local maximum or minimum. - 'f'(x) = -2' at x = c implies the slope of the tangent line is negative, meaning the graph is decreasing at x = c.

Step 3: Examine the graphs provided (a, b, c, d) and identify the one that matches the behavior described in the table. Look for: - A sharp corner or cusp at x = a. - A horizontal tangent line at x = b. - A decreasing slope at x = c.

Step 4: Compare each graph systematically: - Graph (a): Sharp corner at x = a, horizontal tangent at x = b, decreasing slope at x = c. - Graph (b): Smooth curve at x = a, horizontal tangent at x = b, decreasing slope at x = c. - Graph (c): Smooth curve at x = a, horizontal tangent at x = b, increasing slope at x = c. - Graph (d): Sharp corner at x = a, horizontal tangent at x = b, decreasing slope at x = c.

Step 5: Based on the analysis, select the graph that matches all the conditions described in the table. The correct graph is the one with a sharp corner at x = a, a horizontal tangent at x = b, and a decreasing slope at x = c.

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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 occur where the derivative of a function is either zero or undefined. These points are essential for finding local extrema, as they indicate where the function's slope changes, potentially leading to local maxima or minima. In the given table, point 'b' is a critical point since the derivative is zero, while point 'a' is critical because the derivative does not exist.

First Derivative Test

The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can infer the behavior of the function. For instance, if the derivative changes from positive to negative at a critical point, it indicates a local maximum.

Existence of Derivatives

The existence of a derivative at a point is crucial for determining the behavior of a function at that point. If the derivative does not exist, as in point 'a', it may indicate a cusp, corner, or vertical tangent, which can affect the function's continuity and extrema. Understanding where derivatives exist or do not exist helps in analyzing the overall shape and critical features of the graph.

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Related Practice

Textbook Question

110. Suppose the derivative of the function y = f(x) is

y'=(x-1)^22(x-2)(x-4).

At what points, if any, does the graph of f have a local minimum, local maximum, or

point of inflection?

Textbook Question

119. Find the values of constants a, b, and c such that the graph of y = ax^3 + bx^2 + cx has a

local maximum at x = 3, local minimum at x =- 1, and inflection point at (1, 11).

Textbook Question

Finding Extrema from Graphs

In Exercises 11–14, match the table with a graph.

Textbook Question

Find values of a and b such that the function

ƒ(𝓍) = (a𝓍 + b) / 𝓍² ―1)

has a local extreme value of 1 at 𝓍 = 3. Is this extreme value a local maximum or a local minimum? Give reasons for your answer.

Textbook Question

Finding Extreme Values

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

y = 𝓍³ ― 2𝓍 + 4

Textbook Question

Finding Extreme Values

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

y = 𝓍³ + 𝓍² ― 8𝓍 + 5

Textbook Question

101. In Exercises 101 and 102, the graph of f' is given. Determine x-values corresponding to local minima, local maxima, and inflection points for the graph of f.

Multiple Choice

Determine where the local and absolute maxima and minima occur on the given graph of $f\left(x\right)$ .

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Finding Extrema from GraphsIn Exercises 11–14, match the table with a graph.

Key Concepts

Critical Points

First Derivative Test

Existence of Derivatives

Watch next

Finding Extrema from Graphs

In Exercises 11–14, match the table with a graph.