Question

Find local extrema and saddles of f(x,y)=x^3-3xy+y^3? Need help

Answer 1

have you solved \(\nabla f=0\) yet?"

Answer 2

yep you get <3x^2-3y,-3x+3y^2>

Answer 3

then when you set it equal to zero you get y=0 and x=0 right?

Answer 4

there are two solutions, that is one

Answer 5

how do you find the other one?

Answer 6

\[-3x+3y^2=0\implies x^2=y^4\]\[3y^4-3y=0\implies y(y^3-1)=0\implies y=\{0,1\},~x=\{0,1\}\]

Answer 7

?

Answer 8

solve the second equation for x^2 and substitute into the first

Answer 9

so your critical points are (0,0) and (1,1)?

Answer 10

yes

Answer 11

so what do you do with those?

Answer 12

find D for each\[D=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2\]

Answer 13

find second derivatives then plug in the point values to see if they are greater or less than zero?

Answer 14

yes

Answer 15

so you get -9 and 27 so would (0,0) be a local min?