Exercise 9.31

I am going to give the results, but not prove them. You can find a proof in any good advanced calculus text, and it’s easy to find online. (For example, see Theorem 16.4 of Loomis and Sternberg’s Advanced Calculus, which is legally available online now. Actually, why don’t you just read the whole book, it wouldn’t be difficult for you at this point, and it would introduce you to Differential Geometry and Mechanics. They actually used to assign it in advanced Freshman Calculus courses long ago, which must have been a good way to generate a lot of pre-meds.)

For a function $f$ satisfying the above conditions, we have from Exercise 30 that $f$ is approximately the following quadratic function in two variables:

Let $D=(D_{11}f(ǎ))(D_{22}f(ǎ))-(D_{12}f(ǎ){)}^2$. Then the above function will have a local maximum if and only if $D$ is positive and $D_{11}f(ǎ)$ is negative. It will have a local minimum if and only if $D$ is positive and $D_{11}f(ǎ)$ is positive. This will also hold for $f$ near $ǎ$.

For $n$ variables, we need to consider the eigenvalues of the Hessian matrix $(D_{\mathit{ij}}f(ǎ))$. There will be $n$ real eigenvalues since the matrix is symmetric. Then $f$ has a local maximum at $ǎ$ if and only if the eigenvalues are all negative, and it will have a local minimum at $ǎ$ if and only if they are all positive.