SOLUTION: Find two non negative numbers whose sum is 20 and for which the product of twice one number and the square of the other number will be a maximum.

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Question 261753: Find two non negative numbers whose sum is 20 and for which the product of twice one number and the square of the other number will be a maximum.
Answer by drk(1908) About Me  (Show Source):
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Let A and B be the two non negative numbers.
(i) A+%2B+B+=+20
(ii) 2A+%2A+B%5E2+=+max
step 1 - solve for A to get
(iii) A+=+20+-+B
step 2 - substitute (ii) into (i) to get
(iv) 2%2820-B%29+%2A+B%5E2
step 3 - simplified we get
(v) -2B%5E3+%2B+40B%5E2
step 4 - take a derivative and set = 0 to get
(vi) %28-6B%5E2%2B80B%29+=+0
step 5 - factor to get
(vii) %28-2B%283B+-+40%29%29+=+0
We get
B = 0 or B = 13.3333
The closest integer to this is
B =13
so A = 7.
--
7+13 = 20
2*7 * 13^2 = 2366