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).