What are extrema and saddle points of #f(x, y) = x^3y + 36x^2 - 8y#?

Answer 1

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Answer 2

To find the extrema and saddle points of ( f(x, y) = x^3y + 36x^2 - 8y ), follow these steps:

  1. Find the first-order partial derivatives of ( f ) with respect to ( x ) and ( y ):

( f_x = 3x^2y + 72x ) ( f_y = x^3 - 8 )

  1. Set both partial derivatives equal to zero and solve the resulting system of equations to find critical points:

( 3x^2y + 72x = 0 ) ( x^3 - 8 = 0 )

  1. Solve the second equation ( x^3 - 8 = 0 ) to find the possible values of ( x ), then substitute these values into the first equation ( 3x^2y + 72x = 0 ) to find corresponding values of ( y ).

  2. After finding the critical points, determine the nature of each critical point by using the second partial derivative test or the Hessian matrix.

  3. The critical points where the second derivative test confirms a maximum or minimum are extrema. The critical points where the second derivative test results are inconclusive are saddle points.

  4. Calculate the function values at the critical points to confirm whether they are maximum, minimum, or saddle 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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