What are the critical points of #f(x,y) =x^3 + xy - y^3#?

Answer 1

They are #(0,0)# and #(1/3, -1/3)#

For #f(x,y) =x^3 + xy - y^3#, we have
#f_x = 3x^2+y# #f_y = x-3y^2#.

We need to solve the system

#3x^2+y = 0# #x-3y^2 = 0#.
The first equation gives us #y = -3x^2#.
Substituting for #y# in the second equation gets us
#x-3(-3x^2)^2 = 0#
#x-27x^4 = 0#
#x(1-27x^3) = 0#
#x=0# #" "# OR #" "# #x=1/3#.
Using #y = -3x^2# from above we get critical points #(0,0)# and #(1/3, -1/3)#.
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Answer 2

To find the critical points of ( f(x,y) = x^3 + xy - y^3 ), we need to find where the partial derivatives with respect to ( x ) and ( y ) are both zero.

  1. Partial derivative with respect to ( x ): [ \frac{\partial f}{\partial x} = 3x^2 + y ]

  2. Partial derivative with respect to ( y ): [ \frac{\partial f}{\partial y} = x - 3y^2 ]

Setting both partial derivatives equal to zero:

  1. ( 3x^2 + y = 0 )
  2. ( x - 3y^2 = 0 )

Solve these two equations simultaneously to find the critical points.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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