14) Find two positive numbers x and y such that x + y = 60 and xy^3 is maximum.

Answers (1)

It is given that
x + y = 60 , x = 60 -y
and xy^3 is maximum
let f(y) = (60-y)y^3 = 60y^3-y^4
Now,
f^{'}(y) = 180y^2-4y^3\\ f^{'}(y) = 0\\ y^2(180-4y)=0\\ y= 0 \ and \ y = 45

Now,
f^{''}(y) = 360y-12y^2\\ f^{''}(0) = 0\\
hence, 0 is niether point of minima or maxima
f^{''}(y) = 360y-12y^2\\ f^{''}(45) = 360(45)-12(45)^2 = -8100 < 0
Hence, y = 45 is point of  maxima
x = 60 - y
  = 60 - 45 = 15
Hence, values of x and y are 15 and 45 respectively

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