Question
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  1. On the closed interval , the function han
    (A) both an absolute maximum and an absolute minimum.
    (B) an absolute maximum but no absolute minimum.
    © no absolute maximum but an absolute minimum.
    (D) an absolute maximum and two absolute minima.
  2. The critical numbers of the function

Ask by Daniel Chen. in the United States
Jan 22,2025

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Answer

On the interval , the function has both an absolute maximum and an absolute minimum.
Answer:
(A) both an absolute maximum and an absolute minimum.

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Beyond the Answer

The function is shaped like a “V.” It reaches its highest point (the absolute maximum) at , where . As you move away from , the function decreases linearly until it reaches its endpoints, and . Therefore, option (A) is correct: the function has both an absolute maximum and an absolute minimum over the interval .
When finding the critical numbers of , we need to consider where the derivative is zero or undefined. The derivative is piecewise defined because of the absolute value. It’s positive for and negative for , while at , the derivative doesn’t exist due to the sharp corner. Thus, the critical number here is , which is where the maximum occurs, confirming that we found an important point regarding the function’s behavior!

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