The first-order partial derivatives are (partial f)/(partial x)=y-3x^{-4}∂f∂x=y−3x−4 and (partial f)/(partial y)=x-2y^{-3}∂f∂y=x−2y−3. Setting these both equal to zero results in the system y=3/x^(4)y=3x4 and x=2/y^{3}x=2y3. Subtituting the first equation into the second gives x=2/((3/x^{4})^3)=(2x^{12})/27x=2(3x4)3=2x1227. Since x !=0x≠0 in the domain of ff, this results in x^{11}=27/2x11=272 and x=(27/2)^{1/11}x=(272)111 so that y=3/((27/2)^{4/11})=3*(2/27)^{4/11}y=3(272)411=3⋅(227)411
The second-order partial derivatives are (partial^{2} f)/(partial x^{2})=12x^{-5}∂2f∂x2=12x−5, (partial^{2} f)/(partial y^{2})=6y^{-4}∂2f∂y2=6y−4, and (partial^{2} f)/(partial x partial y)=(partial^{2} f)/(partial y partial x)=1∂2f∂x∂y=∂2f∂y∂x=1.
The discriminant is therefore D=(partial^{2} f)/(partial x^{2})*(partial^{2} f)/(partial y^{2})-((partial^{2} f)/(partial x partial y))^{2}=72x^{-5}y^{-4}-1D=∂2f∂x2⋅∂2f∂y2−(∂2f∂x∂y)2=72x−5y−4−1. This is positive at the critical point.
Since the pure (non-mixed) second-order partial derivatives are also positive, it follows that the critical point is a local minimum.
