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Calculus Exam 2 Review: Extrema, Graph Analysis, and Optimization

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Q2. Does f(x) achieve a relative maximum when x = -4?

Background

Topic: Relative Extrema from Graphs

This question tests your ability to identify relative maxima by analyzing the graph of a function. A relative maximum occurs at a point where the function changes from increasing to decreasing.

Graph of f(x) for extrema analysis

Key Terms:

  • Relative Maximum: A point where the function value is higher than all nearby points.

  • Critical Point: A point where the derivative is zero or undefined.

Step-by-Step Guidance

  1. Locate x = -4 on the x-axis of the graph.

  2. Observe the behavior of the function near x = -4: Is the function increasing before x = -4 and decreasing after?

  3. Check if the point at x = -4 is a peak (highest point in its immediate neighborhood).

  4. Consider whether the function is defined at x = -4 (open or closed circle) and if the slope changes sign.

Try solving on your own before revealing the answer!

Final Answer: False

At x = -4, the graph does not show a relative maximum. The function does not change from increasing to decreasing at this point, and the endpoint is not a maximum.

Q3. Is the absolute maximum value of f(x) equal to 4?

Background

Topic: Absolute Maximum from Graphs

This question asks you to determine the highest value the function attains on its domain, based on the graph.

Graph of f(x) for maximum value analysis

Key Terms:

  • Absolute Maximum: The largest value of f(x) over its entire domain.

Step-by-Step Guidance

  1. Scan the graph for the highest point reached by the function.

  2. Check the y-coordinate of this highest point.

  3. Compare this value to 4 to see if it matches.

Try solving on your own before revealing the answer!

Final Answer: True

The graph reaches its highest value at y = 4, so the absolute maximum is 4.

Q4. Is the absolute minimum value of f(x) equal to ?

Background

Topic: Absolute Minimum from Graphs

This question tests your understanding of how to identify the lowest value a function attains, possibly as x approaches infinity or negative infinity.

Graph of f(x) for minimum value analysis

Key Terms:

  • Absolute Minimum: The lowest value of f(x) over its entire domain.

  • End Behavior: The behavior of the function as x approaches or .

Step-by-Step Guidance

  1. Observe the graph as x increases or decreases without bound.

  2. Determine if the function continues to decrease indefinitely (downward arrow) or if it levels off at a certain value.

  3. Check if there is a lowest point or if the function goes to .

Try solving on your own before revealing the answer!

Final Answer: False

The function does not decrease without bound; the lowest value is not based on the graph's behavior.

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