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Question

Feature
The graph of a function with 5 extrema is shown.The extrema are located at left-parenthesis negative 4 comma 3 right-parenthesis, left-parenthesis negative 2 comma negative 2 right-parenthesis, left-parenthesis 0 comma 2 right-parenthesis, left-parenthesis 2 comma negative 3 right-parenthesis, and left-parenthesis 4 comma 4 right-parenthesis. The function approaches negative infinity on both ends.
Question
Use the graph to answer the question.
Which statements about the extrema of the graphed function are true? Select all that apply.
(1 point)
Responses

left parenthesis negative 2 comma negative 2 right parenthesis is a local minimum.
Image with alt text: left parenthesis negative 2 comma negative 2 right parenthesis is a local minimum.

left parenthesis negative 5 comma 0 right parenthesis is a local minimum.
Image with alt text: left parenthesis negative 5 comma 0 right parenthesis is a local minimum.

left parenthesis 4 comma 4 right parenthesis is the global maximum.
Image with alt text: left parenthesis 4 comma 4 right parenthesis is the global maximum.

left parenthesis 2 comma negative 3 right parenthesis is the global minimum.

Answer 1

To determine which statements about the extrema of the graphed function are true, let's analyze the provided extrema:

1. Extrema Points:

   • \( (-4, 3) \)
   • \( (-2, -2) \)
   • \( (0, 2) \)
   • \( (2, -3) \)
   • \( (4, 4) \)

2. Behavior of the Function:

   • The function approaches negative infinity on both ends.

Analysis of Statements:

1. Statement: \( (-2, -2) \) is a local minimum.

   • At \( (-2, -2) \), the function value is lower than the values at nearby extrema \( (-4, 3) \) and \( (0, 2) \). True.

2. Statement: \( (-5, 0) \) is a local minimum.

   • Since \( (-5, 0) \) is not one of the indicated extrema points, we cannot determine its status as a minimum based on the given information about extrema. False.

3. Statement: \( (4, 4) \) is the global maximum.

   • This point has the highest function value compared to the other extrema (as \( (2, -3) \) is lower and approaches negative infinity at both ends). True.

4. Statement: \( (2, -3) \) is the global minimum.

   • While it is the lowest value among the listed extrema, the function approaches negative infinity as \( x \) moves towards the ends, meaning the function has no global minimum in the sense of a highest-lowest value, which would be \( -\infty \). False.

Conclusion:

The true statements are:

• \( (-2, -2) \) is a local minimum.
• \( (4, 4) \) is the global maximum.