Absolute Extrema

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Kristina123

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Hi, I have a couple of questions about this problem.

Over this interval [-π/2,π], the absolute max value would be 2, and the absolute min value would be -0.25. I got my answers through plugging in the critical numbers I found, including the endpoints of the interval, into the function and compared to find the greatest and lowest value. Would that be enough to justify my answer, or is there something else that I'm missing?

Also, suppose I had cos^2x – cos x as my function, wouldn't the absolute max still be 2 and absolute min be -0.25? They have the same parabolic curves over and over again, and it's also continuous, but this time there is no specified closed interval. If there is no closed interval, would there still be absolute extrema?
 

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Kristina123 said:
View attachment 29799
Hi, I have a couple of questions about this problem.

Over this interval [-π/2,π], the absolute max value would be 2, and the absolute min value would be -0.25. I got my answers through plugging in the critical numbers I found, including the endpoints of the interval, into the function and compared to find the greatest and lowest value. Would that be enough to justify my answer, or is there something else that I'm missing?

Also, suppose I had cos^2x – cos x as my function, wouldn't the absolute max still be 2 and absolute min be -0.25? They have the same parabolic curves over and over again, and it's also continuous, but this time there is no specified closed interval. If there is no closed interval, would there still be absolute extrema?
Click to expand...
You didn't miss anything - in my opinion.

I am impressed that you looked at the graph of the function to buttress your answer.
 
Beer inspired graph follows.
Kristina123 said:
View attachment 29799
Hi, I have a couple of questions about this problem.

Over this interval [-π/2,π], the absolute max value would be 2, and the absolute min value would be -0.25. I got my answers through plugging in the critical numbers I found, including the endpoints of the interval, into the function and compared to find the greatest and lowest value. Would that be enough to justify my answer, or is there something else that I'm missing?

Also, suppose I had cos^2x – cos x as my function, wouldn't the absolute max still be 2 and absolute min be -0.25? They have the same parabolic curves over and over again, and it's also continuous, but this time there is no specified closed interval. If there is no closed interval, would there still be absolute extrema?
Click to expand...
www.desmos.com

Kristina123 Absolute Extrema

www.desmos.com www.desmos.com
 
I disagree with what you said. You do not have to plug ALL the critical value into the function. You need plug in all the critical values that lie in your given interval in the function.
 
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