📚 How to find the local maximum/minimum and saddle points of a function (Question 1)
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Q. Find the local maximum or minimum and saddle points of the function f(x,y)=x^3−xy+y^2
In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point.
For a function f of two or more variables, there is a generalization of the rule above. In this context, instead of examining the determinant of the Hessian matrix, one must look at the eigenvalues of the Hessian matrix at the critical point. The following test can be applied at any critical point a for which the Hessian matrix is invertible:
If the Hessian is positive definite (equivalently, has all eigenvalues positive) at a, then f attains a local minimum at a.
If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a.
If the Hessian has both positive and negative eigenvalues then a is a saddle point for f (and in fact this is true even if a is degenerate).
In those cases not listed above, the test is inconclusive.