# Extrema and saddle points

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Examine each function for relative extrema and saddle points:

f (x,y) = x^2 - 3xy + y^2

Â© BrainMass Inc. brainmass.com December 24, 2021, 7:46 pm ad1c9bdddfhttps://brainmass.com/math/graphs-and-functions/extrema-saddle-points-216982

## SOLUTION This solution is **FREE** courtesy of BrainMass!

We are given f(x, y) = x^2 - 3xy + y^2.

We calculate

f_x = 2x - 3y

f_y = -3x + 2y

so the critical points (a, b) are solutions to

2a - 3b = 0

-3a + 2b = 0

in which case (a, b) = (0, 0) is the only solution.

We also have

f_xx = 2

f_yy = 2

f_xy = -3

D := (f_x)(f_y) - (f_xy)^2 = (2x - 3y)(-3x + 2y) - 9

so that D(0, 0) = - 9 < 0, in which case f does *not* have a relative extremum

at (0, 0).

https://brainmass.com/math/graphs-and-functions/extrema-saddle-points-216982