Proving the Second Partials Test

[mattmns]

Hello. I am just wondering about how to start this problem: Prove the Second Partials Test. The book I am using gives a hint of: Compute the directional derivative of f in the direction of the unit vector u = hi + kj and complete the square.

I guess I am just having trouble even starting it. Am I supposed to define f or something of that nature to start? I just cannot see how to take the derivative of f without knowing what f is. Thanks.

edit... Well I thought about it, and I guess I could say that the directional derivative is $$f_x h + f_y k$$

But completing the square there? Do I need to take the directional derivative again? $$(f_{xx} h + f_{yx} k)h + (f_{xy} h + f_{yy} k)k$$ Then complete the square?

 

[Nate810]

Thanks. Yes, you are correct. To prove the second partials test, you need to compute the directional derivative of f in the direction of the unit vector u = hi + kj, and then complete the square by factoring out the h^2 and k^2 terms. The result should be a quadratic in h and k which can then be evaluated to determine if it is positive, negative, or zero.

 

[thepatientmental]

To start the problem, you would need to define the function f first. The Second Partials Test is a theorem that helps determine the nature of a critical point of a function of two variables. So, you would need to have a function of two variables, let's say f(x,y), to apply this test.

The hint given in the book is suggesting that you compute the directional derivative of f in the direction of the unit vector u = hi + kj, which is the vector <h,k>. This can be written as f_x h + f_y k, where f_x and f_y are the partial derivatives of f with respect to x and y, respectively. This is the general formula for the directional derivative in the direction of a vector <h,k>.

Now, to complete the square, you would need to rewrite the directional derivative in a specific form. Since the Second Partials Test involves the second partial derivatives of f, we need to have terms in the form of f_xx, f_xy, and f_yy. So, you would need to rearrange the terms in the directional derivative to get something like f_xx h^2 + f_xy hk + f_yy k^2. This can be done by multiplying the original directional derivative by h/h and k/k, respectively, and then grouping the terms.

Once you have the directional derivative in this form, you can complete the square by adding and subtracting the term (f_xy)^2. This will give you a perfect square of the form (f_xx h + f_xy k)^2 + (f_yy k)^2. You can then factor out the common term of (f_xx h + f_xy k)^2 and rewrite the expression as (f_xx h + f_xy k)^2 + (f_yy k)^2 - (f_xy)^2.

This is the completed square form of the directional derivative, and it can be used to determine the nature of a critical point of f. The Second Partials Test states that if the directional derivative is positive at a critical point, then the point is a local minimum; if it is negative, then the point is a local maximum; and if it is zero, then the test is inconclusive.

So, to prove the Second Partials Test, you would need to show that the completed square form of the directional derivative can be used to determine the nature of a critical point of f. You can do