extremum problem

one surface as follows:

f(x,y)=ax2+by2+cxy+dx+ey+g

I wish find $ (x^{*},y^{*})=argmin f(x,y) $

then:

f_x^{'}(x,y)=2ax+cy+d

f_y^{'}(x,y)=cx+2by+e

Set these two equations to zero and solve to find the extremum. Once you have it you can test if it is max, min, inflection or unkown by finding D:

D=f_{xx}(a,b)f_{yy}(a,b)-f_{xy}^{2}(a,b)=4ab-c^2

Then, if:

D>0\, and\, f_{xx}(a,b)>0\, then\, f\, is\, min\, at\, (a,b)

D>0\, and\, f_{xx}(a,b)<0\, then\, f\, is\, max\, at\, (a,b)

D<0\, then\, f\, is\, a\, saddle\, point\, at\, (a,b)

D=0\, no\, conclusion

 

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