Differentiation done correctly: 5. Maxima and minima

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima

In this final post, we are going to look at some applications of differentiation to locating maxima and minima of real valued functions. In order to do this, we will be using Taylor’s theorem (covered in part 2) to prove the higher derivative test for functions on Banach spaces, and the implicit function theorem (covered in part 4) to prove a special case of the method of Lagrange multipliers.

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Differentiation done correctly: 3. Partial derivatives

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima

While we saw that differentiable maps may be naturally split into component functions when the codomain is a product of Banach spaces, the situation for the domain is more complicated. (This is partly due to the fact that as a topological space, there is no natural injection into a product of Banach spaces.) In this post, we will look at how the existence of partial derivatives relates to differentiability, how the symmetry of higher derivatives (covered in part 2) affects mixed partial derivatives, and finally a short proof of differentiation under the integral sign.

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Differentiation done correctly: 2. Higher derivatives

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima

Last time, we covered the definition of the derivative and its basic properties, which all turn out to be quite similar to their single variable counterparts. Now we are going to explore higher derivatives. In traditional multivariable calculus, true higher derivatives do not exist (except in a specific situation which will be discussed in part 5). Of course, we have so-called “mixed/higher partial derivatives”, which are coordinate-dependent and notationally tricky to work with. As a consequence, the usual statement of Taylor’s theorem in \(\mathbb{R}^n\) ends up being ugly and hard to remember. In reality, Taylor’s theorem for Banach spaces looks almost exactly the same as the single variable Taylor’s theorem!

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Every continuous open mapping of R into R is monotonic

I’m willing to bet that most students who have used Rudin’s Principles of Mathematical Analysis have encountered this problem:

15. Call a mapping of \(X\) into \(Y\) open if \(f(V)\) is an open set in \(Y\) whenever \(V\) is an open set in \(X\).

Prove that every continuous open mapping of \(\mathbb{R}\) into \(\mathbb{R}\) is monotonic.

Obvious?

Here is one “solution” that is fairly intuitive. It relies on finding a minimum or maximum and considering the image of a small neighborhood around that min/max:

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PAE patch updated for Windows 8

Note: An updated version for Windows 8.1 is available.

This patch allows you to use more than 3/4GB of RAM on an x86 Windows system. Works on Vista, 7, 8, has been tested on Windows Vista SP2, Windows 7 SP0, Windows 7 SP1 and Windows 8 SP0. Instructions and source code included.

Download: PatchPae2.zip (346094 downloads)

Before using this patch, make sure you have fully removed any other “RAM patches” you may have used. This patch does NOT enable test signing mode and does NOT add any watermarks.

Note: I do not offer any support for this. If this did not work for you, either:

  • You cannot follow instructions correctly, or
  • You cannot use more than 4GB of physical memory on 32-bit Windows due to hardware/software conflicts. See the comments on this page for more information.

Free product of free groups and group presentations

Here is Problem 9-4(b) from Introduction to Topological Manifolds by John M. Lee:

Let \(S_1\) and \(S_2\) be disjoint sets, and let \(R_i\) be a subset of the free group \(F(S_i)\) for \(i=1,2\). Prove that \(\langle S_1 \cup S_2 \mid R_1 \cup R_2 \rangle\) is a presentation of the free product group \(\langle S_1 \mid R_1 \rangle * \langle S_2 \mid R_2 \rangle\).

The proof is fairly straightforward if we stick to the universal properties of the free group, free product and quotient group. Here I will write out the full details of the proof, which is more fun to do than to read. There are three basic steps to showing that two things are isomorphic:

  1. Set up the canonical maps and choose names for them. These maps are given to you by the definition of the free group, free product, etc.
  2. Repeatedly use the existence part of the universal properties to derive two maps.
  3. Use the uniqueness part of the universal properties to show that the maps are inverses to each other.

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