Home page for accesible maths 4 Chapter 4 contents 4.9 Classification of stationary points via the discriminant 4.11 Discriminant test

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4.10 Examples of the Hessian discriminant

Example.

Find the stationary points of $f(x,y)=x^{2}+xy-2y^{2}-2$ and calculate the Hessian discriminant at each stationary point.

Solution. We first calculate partial derivatives:

f_{x}=\,{2x+y,}\;\;f_{y}=\,{x-4y,}\;\;f_{xx}=\,{2,}\;\;f_{xy}=\,{1,}\;\;f_{yy}% =\,{-4.}

So the stationary points are given by $2x+y=x-4y=0$ , whence $x=4y$ and $y=-2x=-8y$ so we get $x=y=0$ . Thus there is only one stationary point: $(0,0)$ .

Now the Hessian discriminant is

f_{xx}f_{yy}-f_{xy}^{2}=\,{2.(-4)-1^{2}=}\,{-9.}