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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 = 35-x

product = x^2y^5 = x^2(35-x)^5
d(product)/dx = x^2(5)(35-x)^4 (-1) + 2x(35-x)^5
= 0 for a max of product

x^2(5)(35-x)^4 (-1) + 2x(35-x)^5 = 0
x(35-x)^4 [ -5x + 2(35-x) ] = 0
x(35-x)^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 = 35-10 = 25
The two numbers are 10 and 25

55%

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