Math Problem Statement

The graph of a function f is given. Use the graph to find each of the following. a. The​ numbers, if​ any, at which f has a relative maximum. What are these relative​ maxima? b. The​ numbers, if​ any, at which f has a relative minimum. What are these relative​ minima? -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y

A coordinate system has a horizontal x-axis labeled from negative 6 to 6 in increments of 1 and a vertical y-axis labeled from negative 6 to 6 in increments of 1. From left to right, a curve falls at a decreasing rate and comes to a minimum at (2, negative 1). The curve then rises, first at an increasing rate, and then at a decreasing rate until it reaches a maximum at (3, 3). The curve then falls, first at an increasing rate, and then at a decreasing rate until it reaches a minimum at (4, negative 1). The curve then rises at an increasing rate for the rest of the graph. Question content area bottom Part 1 ​(a) Find the​ numbers, if​ any, at which f has a relative maximum. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The ​number(s) at which f has a relative maximum​ is/are enter your response here. ​(Type an integer or a decimal. Use a comma to separate answers as​ needed.) B. There is no number at which f has a relative maximum.

Solution

Given the description of the graph, we can analyze the function f(x)f(x) to determine the points at which it has relative maxima and minima.

(a) Relative Maximum:

A relative maximum occurs at a point where the function changes direction from increasing to decreasing. From the description:

  • The function rises after reaching a minimum at (2,−1)(2, -1) and continues to rise until it reaches (3,3)(3, 3), where it then starts to decrease.

This indicates that f(x)f(x) has a relative maximum at x=3x = 3, with the value of the maximum being f(3)=3f(3) = 3.

Answer:

A. The number at which ff has a relative maximum is x=3x = 3, and the value of the relative maximum is 3.

(b) Relative Minimum:

A relative minimum occurs at a point where the function changes direction from decreasing to increasing. From the description:

  • The function decreases to a minimum at (2,−1)(2, -1), rises to a maximum at (3,3)(3, 3), then decreases again to another minimum at (4,−1)(4, -1).

This indicates that f(x)f(x) has relative minima at x=2x = 2 and x=4x = 4, with the value of the minima being f(2)=−1f(2) = -1 and f(4)=−1f(4) = -1.

Answer:

The numbers at which ff has relative minima are x=2x = 2 and x=4x = 4, and the value of each relative minimum is −1-1.

Do you have any further questions, or would you like more details on this explanation?

Related Questions:

  1. What is the general definition of a relative maximum and minimum?
  2. How would you find relative extrema of a function using derivatives?
  3. Can a function have more than one relative maximum or minimum? Explain.
  4. What are the differences between absolute and relative extrema?
  5. How can the second derivative test help determine if a critical point is a maximum or minimum?

Tip:

To determine relative maxima and minima, always look for points where the slope of the tangent line to the curve changes sign, indicating a change in the direction of the graph.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Functions
Relative Extrema

Formulas

-

Theorems

First Derivative Test

Suitable Grade Level

Grades 11-12

Related Recommendation

Finding Relative Maxima and Minima from a Graph

Identify Relative Extrema from a Graph

Identify Local Maxima and Minima in Graphs - Step-by-Step Guide

Identifying Local Maxima and Minima from a Graph

Analyzing a Function Graph to Find Absolute Maximum and Minimum