Question

Locate and classify the stationary points of the function

f( x,y ) x^2+y^2–6xy +6x +3y –4

Answer 1

$f_x'=2x-6y+6$ ,$f_y'=2y-6x+3$ , this implies $x=\frac{15}{16},y=\frac{21}{16}$ is a stationary point.

$f_{xx}''=2$,$f_{xy}''=-6$ ,$f_{xy}''=2$ hence $(f_{xy}'')^2-(f_{xx}'')(f_{yy}'')^2>0$, hence there is no extremum at this point, it is a saddle point.