# Thread: Help with Critical Points

1. ## Help with Critical Points

Hey could someone take me through the steps of finding the critical points of the following function:

f(x,y) = (x^2y^4)/4 + 4x^2 - xy^3 + y^2

Thankyou!

2. The critical points of a function are the points at which the partial derivatives are 0 or the derivatives do not exist.

This function, $\displaystyle f(x,y) = (x^2y^4)/4 + 4x^2 - xy^3 + y^2$, is a polynomial in x and y so it can be done using the fact that the derivative of $\displaystyle x^n$, with respect to x, is $\displaystyle nx^{n-1}$ and the derivative of $\displaystyle y^n$, with respect to y, is $\displaystyle ny^{n-1}$. Where are you having difficulty? Have you found the partial derivative functions?

3. Yes so i have managed to do this

fx = (xy^4)/2 + 8x - y^2

0 = x(1/2y^4 + 8 - y^3) .... 1.

fy = x^2 - 3xy^2 +2y

0 = y(x^2y^2 - 3xy +2) ...... 2.

Eqn 1. x = 0 or 1/2y^4 + 8 - y^3 = 0

with x = 0 Eqn 2. becomes 2y = 0 so y = 0
therefore first critical point = (0,0)

with 1/2y^4 + 8 - y^3 = 0

And now i'm stuck here. Have I done the right thing so far? I am really confused because i'm trying to following an example in my text book but its a much simpler function than my own.