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!
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?
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.