Sufficient conditions for a critical point of a smooth function f (x,y,z)

[Win_odd Dhamnekar]

The following theorem gives sufficient conditions for a critical point to be a local maximum or minimum of a smooth function (i.e. a function whose partial derivatives of all orders exist and are continuous).


What are the changes we, must make in this theorem in case of f(x,y,z) and f( w, x, y, z)?

How can we use the aforesaid rectified theorem to answer the following question?

Find three positive numbers x, y, z whose sum is 10 such that x²*y²*z is a maximum.

My attempt : The critical point is x=4, y=4, z =2

How to compute D in this case?

 

[BigBeachBanana]

The following theorem gives sufficient conditions for a critical point to be a local maximum or minimum of a smooth function (i.e. a function whose partial derivatives of all orders exist and are continuous).

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What are the changes we, must make in this theorem in case of f(x,y,z) and f( w, x, y, z)?

How can we use the aforesaid rectified theorem to answer the following question?

Find three positive numbers x, y, z whose sum is 10 such that x²*y²*z is a maximum.

My attempt : The critical point is x=4, y=4, z =2

How to compute D in this case?

You're optimizing [imath]f(x,y)=x^2y^2(10-x-y)[/imath]

 

[blamocur]

The theorem in your post is about the Jacobian determinant for the 2D case. You can read about the general case at, among other places, https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant.