math 12 grade
posted by ramesh .
find two positive number X and Y such that their sum is 35 and the product X2 Y5 is maximum?

did you mean x^2 y^5 ???
I will assume that.
x+y = 35 > y = 35x
product = x^2y^5 = x^2(35x)^5
d(product)/dx = x^2(5)(35x)^4 (1) + 2x(35x)^5
= 0 for a max of product
x^2(5)(35x)^4 (1) + 2x(35x)^5 = 0
x(35x)^4 [ 5x + 2(35x) ] = 0
x(35x)^4 [ 7x + 70 ] = 0
x = 0 or 7x + 70 = 0
so x = 0 or x = 10 , but we wanted positive numbers
so x = 10 and y = 3510 = 25
The two numbers are 10 and 25 
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