Question
- 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.
- The critical numbers of the function
(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.
Ask by Daniel Chen. in the United States
Jan 22,2025
Upstudy AI Solution
Tutor-Verified Answer
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.
(A) both an absolute maximum and an absolute minimum.
Solution

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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!