What are the critical points of # f(x,y)=x^3 y + 4x^2 − 8y#?

Answer 1

Samantha Adkison

#(2, -4/3)#

#f_x(x,y) = 3x^2y+8x#

#f_y(x,y) = x^3-8#

Solve: #3x^2y+8x = 0#

#x^3-8 = 0# #" "# so #x=2# now substitute in the other equation to get

#12y+16 = 0# #" "# so #y=-4/3#

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Answer 2

Samantha Adkison

To find the critical points of ( f(x,y) = x^3 y + 4x^2 - 8y ), we need to find where the partial derivatives with respect to ( x ) and ( y ) are both equal to zero.

Find ( \frac{\partial f}{\partial x} ):

[ \frac{\partial f}{\partial x} = 3x^2 y + 8x ]

Find ( \frac{\partial f}{\partial y} ):

[ \frac{\partial f}{\partial y} = x^3 - 8 ]

Set both partial derivatives equal to zero and solve for ( x ) and ( y ) separately:

[ 3x^2 y + 8x = 0 ] [ x^3 - 8 = 0 ]

Solving these equations will give us the critical points. Once we have these points, we can verify whether they are maximums, minimums, or saddle points by using the second derivative test or other relevant methods.

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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.