15) Find two positive numbers x and y such that their sum is 35 and the product x^2 y^5
is a maximum.

Answers (1)

It is given that
x + y = 35 , x = 35 - y
and x^2 y^5 is maximum
Therefore,
let \ f (y )= (35-y)^2y^5\\ = (1225-70y+y^2)y^5\\ f(y)=1225y^5-70y^6+y^7
Now,
f^{'}(y) = 6125y^4-420y^5+7y^6\\ f^{'}(y)=0\\ y^4(6125-420y+7y^2) = 0 \\y =0 \ and \ (y-25)(y-35)\Rightarrow y = 25 , y=35
Now,
f^{''}(y)= 24500y^3-2100y^4+42y^5

f^{''}(35)= 24500(35)^3-2100(35)^4+42(35)^5\\ = 105043750 > 0
Hence, y = 35 is the point of minima

f^{''}(0)= 0\\
Hence, y= 0 is neither point of maxima or minima

f^{''}(25)= 24500(25)^3-2100(25)^4+42(25)^5\\ = -27343750 < 0
Hence, y = 25 is the point of maxima
x = 35 - y
   = 35 - 25 = 10 
Hence, the value of x and y are 10 and 25 respectively

 

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