11.7#6

Hello. I found the derivative with respect to x which is [ y-4xy-5y^2] and the derivative with respect to y which is [ x-10xy-2x^2]. I then used those equations to find the critical points. I got that x=0,x=1/2 and y=0,y=1/5. So using a combination of these I got three out of the four critical points, but I can't get the fourth from what I have.

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Well, the derivative with respect to x is (1 - 4 x - 5 y) y and the derivative with respect to y is
x(1-2x-10y) so you have the equatoins (1 - 4 x - 5 y) y=0 and x(1-2x-10y)=0. The first equation can be solved if either y=0 or (1 - 4 x - 5 y)=0 and the second equation if  either x=0 or (1-2x-10y) =0. One of each pair must be true at the same point (x,y), so that the options are (x=0,y=0)-->(0,0), or
(x=0, 1-4x-5y=0)-->(0,1/5), or (1-2x-10y=0, y=0)-->(1/2,0) or
(1-2x-10y=0, 1-4x-5y=0)-->(1/6, 1/15).