Question

Examine the function for relative extrema and saddle points. (if an answer does not exist, enter dne.) h(x, y) = x2 − 9xy − y2

Answer 1

The given function is
h(x,y) = x² - 9xy - y²

Calculate partial derivatives.
$h_{x} = 2x \\ h_{xx} = 2 \\ h_{y} = -2y \\ h_{yy} = -2 \\ h_{xy} = 0$
For the critical points,
$h_{x} = 2x=0 \, \Rightarrow \, x=0 \\h_{y}=-2y=0 \,, \Rightarrow \, y=0$

To perform a test for a saddle point, calculate
$D = h_{xx}(0,0) h_{yy} (0,0) -[ h_{xy}(0,0)]^{2} = (2)(-2) - (2)^{2} = -8$
Because D < 0, a saddle point exists at (0,0).