For $f(x,y) = x^2 - y^2$ , the origin is a critical point. $f_{xx} = 2$ , $f_{yy} = -2$ , $f_{xy} = 0$ . The determinant is:

Hints

$D = f_{xx} f_{yy} - f_{xy}^2$ .
Negative $D$ means saddle.

Show the answer

B. $D = -4$ : saddle point

Why

$D = 2 \times (-2) - 0 = -4 < 0$ : saddle point. The function rises along the $x$ -axis and falls along the $y$ -axis, a classic hyperbolic paraboloid.

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For f(x,y)=x2−y2f(x,y) = x^2 - y^2f(x,y)=x2−y2, the origin is a critical point. fxx=2f_{xx} = 2fxx​=2, fyy=−2f_{yy} = -2fyy​=−2, fxy=0f_{xy} = 0fxy​=0. The determinant is:

Practise Unconstrained Multivariable Extrema

More questions on Unconstrained Multivariable Extrema

For $f(x,y) = x^2 - y^2$ , the origin is a critical point. $f_{xx} = 2$ , $f_{yy} = -2$ , $f_{xy} = 0$ . The determinant is: