Math Problem Statement

Use the graph to identify the points on the interval left bracket a comma b right bracket at which local and absolute extreme values occur. Question content area bottom left Part 1 List the​ x-values where local minima​ occur, if any. Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice. A. The local minimum​ value(s) occur(s) at xequals    q comma s. ​(Use a comma to separate answers as​ needed.) Your answer is correct.B. There is no local minimum. Part 2 List the​ x-values where local maxima​ occur, if any. Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice. A. The local maximum​ value(s) occur(s) at xequals    r. ​(Use a comma to separate answers as​ needed.) Your answer is correct.B. There is no local maximum. Part 3 List the​ x-values where absolute minima​ occur, if any. Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice. A. The absolute minimum​ value(s) occur(s) at xequals    enter your response here. ​(Use a comma to separate answers as​ needed.) B. There is no absolute minimum. . . . Question content area right Part 1 a q r s b x y y equals f left parenthesis x right parenthesis

A coordinate system has an unlabeled vertical y-axis and a horizontal x-axis with the following labeled values, listed from left to right: a, q, r, s, and b. A curve labeled y equals f left parenthesis x right parenthesis starts above the x-axis at a and falls from left to right until reaching q. Then the curve rises from left to right until reaching r, which is below the previous location of the curve at a. Next, the curve falls from left to right until reaching s. At s, the curve is below the previous location of the curve at q. The curve has a cusp at s. Finally, the curve rises from left to right until ending at b. At b, the curve is still below the previous location of the curve at q. There are vertical dashed lines extending from the labeled values on the x-axis to the curve.

Solution

To solve this problem, we'll analyze the graph's behavior at the points aa, qq, rr, ss, and bb. Based on the description provided, here’s the step-by-step process to identify local and absolute extrema:

Part 1: Local Minima

A local minimum occurs when the graph changes direction from decreasing to increasing. According to the description:

  • The curve falls from aa to qq, indicating a decrease.
  • Then it rises from qq to rr, indicating an increase.
  • The graph falls again from rr to ss, and since it has a cusp at ss, this suggests a sharp turn.

Thus, qq is a local minimum because the graph goes from decreasing to increasing at this point. No other point shows this type of behavior. Therefore, the correct answer is:

A. The local minimum values occur at x=qx = q.

Part 2: Local Maxima

A local maximum occurs when the graph changes direction from increasing to decreasing. Based on the description:

  • The graph rises from qq to rr, then falls from rr to ss.

Thus, rr is a local maximum because the graph changes from increasing to decreasing. Therefore, the correct answer is:

A. The local maximum values occur at x=rx = r.

Part 3: Absolute Minima

An absolute minimum is the lowest point on the entire interval [a,b][a, b]. The key point to identify is the position of ss relative to the other points on the graph:

  • ss is lower than qq, and the graph has a cusp at ss, meaning it is a sharp turn.
  • The curve rises after ss but ends below the previous height at qq, suggesting ss is the lowest point in the interval.

Thus, the absolute minimum occurs at ss. Therefore, the correct answer is:

A. The absolute minimum value occurs at x=sx = s.

Summary:

  • Local minima: x=qx = q
  • Local maxima: x=rx = r
  • Absolute minima: x=sx = s

Do you want more details on extrema or have any other questions?

Here are some related questions to further understand this concept:

  1. How do you determine if a point is a local maximum or minimum?
  2. Can a cusp be an extremum point?
  3. What is the difference between local and absolute extrema?
  4. How do absolute extrema differ from endpoints?
  5. Why are second derivatives useful in identifying concavity and extrema?

Tip: Always check the behavior of the function before and after a point to determine whether it’s a local extremum.

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Math Problem Analysis

Mathematical Concepts

Calculus
Graph Analysis
Extrema
Intervals

Formulas

-

Theorems

Extreme Value Theorem
Local Extrema Identification

Suitable Grade Level

Grades 11-12

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