How to find whether point is maxima/minima/saddle point for this case?

$\begingroup$

I read that we have a formula to check signs of

$f_{xx}*f_{yy}-(f_{xy})^2$ and $f_{xx}$

And then we conclude whether the point is a maxima minima or saddle point for the given two variable function $f_{(x,y)}$

The text also states that for

$f_{xx}*f_{yy}-(f_{xy})^2$ $=0$ The test is inconclusive. So, how do we proceed for such points in the following example :

$f_{(x,y)}=2(x-y)^2-x^4-y^4$ at $(0,0)$ check maxima/minima/saddle point

$\endgroup$

1 Answer

$\begingroup$

Well, if the second derivative test fails (i.e. the determinant of the Hessian matrix is zero at that point)

We need to check by definition, we take a neighborhood of the point $(0,0)$ using some variables $h,k$, if the point $f(0+h, 0+k) - f(0,0) $ has a certain sign we know it is an extremum, if it is positive for all $h,k$ then we know it's maxima, negative minima, and if the sign is not known and depends on the variables then it's a saddle point.

$$ f(0+h, 0+k) -f(0,0) = f(h,k) -0 = 2(h-k)^2 -h^4 - k^4 $$

We can clearly choose some values for $h,k$ that would return a positive value, and some that return negative values, thus it is a saddle point.

$\endgroup$ 2

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy