Locate all the critical points of the function f(x,y) = 10x - x^2 - 5xy^2

Question

Locate all the critical points of the function f(x,y) = 10x - x^2 - 5xy^2

I got the critical point (0, sqrt2) and (5, 0), But I can't figure out what the third critical point is.

Answer 1

f(x,y) = 10x - x² - 5xy²

∂f/∂x = 10 - 2x - 5y² = 0

∂f/∂y = -10xy = 0

Notice that the bottom equation -10xy = 0 means that both x and y have to be 0, so this can be used to help solve for the top equation:

When x=0:

10 - 2x - 5y² = 0

10 - 2(0) - 5y² = 0

10 - 5y² = 0

10 = 5y²

2 = y²

±√2 = y

(0,√2) and (0,-√2)

When y=0:

10 - 2x - 5y² = 0

10 - 2x - 5(0)² = 0

10 - 2x = 0

10 = 2x

5 = x

(5,0)

Thus, the critical points of the multivariable function are (0,√2), (0,-√2), and (5,0). Looks like your mistake was forgetting that -√2 works as a solution for y because squaring a negative number nets you a positive.

Hope this helped!