What is the meaning of the Fxy (or Fyx) second partial derivative of a multivariable equation? Applications?

Question

I don't understand what the point of taking the Fxy (or Fyx) 2nd partial derivative. I can understand taking Fxx or Fyy, but Fxy seems pointless.

Answer 1

well,
since you have 2 directions, x and y,
when considering the second derivative, it is important to know
how the derivatives f_x and f_y vary with respect to x and y.

so f_xy = partial derivative with respect to x of (f_y)
and f_yx= partial derivative with respect to y of f_x

Answer 2

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RE:
What is the meaning of the Fxy (or Fyx) second partial derivative of a multivariable equation? Applications?
I don't understand what the point of taking the Fxy (or Fyx) 2nd partial derivative. I can understand taking Fxx or Fyy, but Fxy seems pointless.

Answer 3

You may recall from single-variable calculus that when you find a critical point of a function, you can determine if it corresponds to a local maximum or minimum by examining the second derivative at that point, if it exists. If it's positive, then the function has a local minimum at that point, and if it's negative, then it has a local maximum.

The generalization to multivariable functions is this: if the gradient of a function is equal to the zero vector at a point, then that point is a critical point. To determine if the function has a local maxmium or minimum at that point, you examine the matrix consisting of ALL second partial derivatives. If the matrix is positive definite (which, without getting into the linear algebra behind this, is true if Fxx*Fyy - Fxy^2, Fxx and Fyy are all positive), then the critical point is a local minimum. If Fxx, Fyy are negative and Fxx*Fyy-Fxy^2 is positive, it's a local maximum. Otherwise, it's a saddle point.

Answer 4

how to solve it ??????