Question

find the critical points of f(x y) = 4+x^3+y^3 -3xy

Answer 1

the critical points are where the differential of f(x y) = 0. so you want to know where the partial of f with respect to x and the partial of f with respect to y are 0. In other words you want to solve the system of equations:

\[0 =3x^2 - 3y\]
\[0 = 3y^2 - 3x\]

Answer 2

\[x(1-x ^{-3/4})=0 and y(1-y ^{-3/4})\]

Answer 3

so x=0 or x=1 and y=0 or y=1 then what

Answer 4

you see which values correspond to which by plugging either x or y into one of the equations and the points you get are the answers.

Answer 5

sorry, then you have to plug those points into the original function f(x,y) and THAT is your answer.

Answer 6

so in this case you would plug in (0,0) and (1,1) and you end up with the critical points (0,0,4) and (1,1,1)

Answer 7

so why wouldn't we use (0,1) or (1,0)

Answer 8

well you would plug say x = 0 into one of the equations in the partial derivatives. so in the equation

\[0 = 3x^2 - 3y\]

if x =0 then y = 0

Answer 9

oh thanks wish i could give you too medals

Answer 10

haha thanks. and no problem.