極大極小值的數學題問題

lulu

2009-08-18 8:02 am

數學題:
將24m之繩子分成二段,一段圍成正方形,另一段圍成圓形,問應如何分段,才能使面積和為最大?

書中答案為24m/4+pi圍成圓形,其餘為正方形.

答案我是可以算到的,但按照求極大極小的公式,二階導函為小於0時才有極大值,但我二階導數後,發現是大於0,即為極小值....請問,我算錯了嗎??????????????

回答 (1)

用戶10012

2009-08-18 3:00 pm

✔ 最佳答案

You are correct, 24pi/(4 + pi) is a minimum, not a maximum, so this is not the answer ( don't understand why the book said this is the answer ??)
Consider the following:
(1) 24 m is used totally to make a circle, radius = 12/pi, so area = pi(12/pi)^2 = 144/pi = 45.8 m^2.
(2) 24 m is used totally to make a square, side = 24/4 = 6, area = 36.
(3) 24pi/(4 + pi) is used to make a circle, balance to make a square, total area = 20.2.
so case (1) is the answer.
When facing problems like this one, you have to consider the end- points for correct answer.

2009-08-18 07:12:20 補充：
To find the end - points, put x = 0 and x = 24 into the equation for area.

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