I am studying the second partial derivative test and it says when determinant of the hessian matrix is $0$ then there is no conclusion. However I saw on a website a method that tells you how to workout what kind of point is the one with zero determinant. Given an initial function $f$, it says to write a new function $f_1=f-c$, where $c=f(x_0,y_0)$. Then, the rule states what follows. If there is a neighborhood of $(x_0,y_0)$ where $f_1$ is negative in every point except $(x_0,y_0)$ then $(x_0,y_0)$ is a local maximum. If there is a neighborhood of $(x_0,y_0)$ where $f_1$ is positive in every point except$(x_0,y_0)$ then $(x_0,y_0)$ is a local minimum. Lastly, if in every neighborhood of $(x_0,y_0)$ $f_1$ has both positive and negative values then $(x_0,y_0)$ is a saddle point. However, I couldn't verify if the method is correct, is there a proof of this result?
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$\begingroup$No proof needed, since that's not a “rule”, it's the definition of what the words “(strict) local maximum” (etc.) mean!
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