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Question

Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.

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Solution

We have to maxims x2y5 and we have x+y=35

So put x=35y in x2y5

we get (1225+y270y)(y5)

f(y)=1225y5+y770y6

f(y)=6125y4+7y6420y5

Putting this to zero we get
y=35,y=25

f′′(y)=24500y3+42y52100y4
f′′(35)=24500×353+42×3552100×354
f′′(35)>0
At y=25
f′′(25)=24500×253+42×2552100×254
f′′(25)<0
So given function will have maxima when y=25

So x=10

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