consider two nonnegative numbers x and y such that x+y=10. Maximize and minimize the quantities.

Question

x^2y^2

please do it step by step becouse I dont understand this topic

Answer 1

I am assuming you want to find the extrema of x^2 * y^2

x^2 * y^2 = x^2 (10-x)^2 = x^2 (100 - 20x + x^2) = x^4 - 20x^3 + 100x^2 = f(x)

Then

0=f'(x) = 4x^3 - 60x^2 + 200x

0 = x (4x^2 - 60x + 200)

= 4x( x^2 - 15x + 50)

= 4x( x - 10 )(x - 5 )

x = 0 , x = 10 , x = 5

so the extrema are (x=0, y=10)

(x=10, y = 0)

(x=5,y=5)

the min is 0 and the max is 625