This is what I have so far. Please help, I cannot figure out how to find the cri

Question

This is what I have so far. Please help, I cannot figure out how to find the critical points and after that.

1) Find the absolute maximum and absolute minimum of f(x,y)=x4y2-4m + y the region R = (x,y):-25 x s 2,-25 y and identify the x-and y-coordinates of the points where they occur. Be sure to identify all the critical points on the boundary of the region. Also, for any critical points in the interior of the region, indicate whether they are relative maxima, relative minima, or saddle points.

Explanation / Answer

4x^3y^2 - 4y = 0
x^3y^2 - y = 0
y(x^3y - 1) = 0
y = 0 , x^3y = 1 ---> equatiom 1

2x^4y - 4x + 3y^2 = 0 ---> equation 2

Now, put y = 0 into eqn 2 :
-4x = 0
x = 0

Now, put x^3y = 1 into equation 2 :
2x - 4x + 3/x^6 = 0
-2x + 3/x^6 = 0
(3 - 2x^7)/x^6 = 0
x = (3/2)^(1/7)

When x = (3/2)^(1/7),
y = 1/x^3, which becomes
y = (2/3)^(3/7)

So, we have :
criticals
(0,0)
((3/2)^(1/7) , (2/3)^(3/7))

Hope this made sense !

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