质疑现状

An Example

If you have taken a calculus course, you have likely seen a type of exercise called an optimization problem. They go something like this:

Bob has to fence off a rectangular plot with 500 feet of fence. What is the largest area he can enclose?

The statement is usually specially contrived such that it can be done by making a function, taking its derivative, and finding a zero in it to maximize the function.

Let $x$ be one of the side lengths of the rectangle, and since the opposite sides add up to $250$, the other side length is $250-x$. Then we just have to find the maximum of $f(x)=x(250-x)$.

The derivative of $f$ is $250-2x$, which has a zero at $x=125$. So the function is maximized at $x=125$. Plugging stuff in, we find that $f(125)=125\times 125= 15625\text{ ft}^2$.

Another way

But having already done competition math for a few years, I knew about another method used in competitions to dispatch such types of problems. (The method is called AM-GM, which basically says that the arithmetic mean is never less than the geometric mean.)

So I was able to deal with them much more quickly than everyone else, who continued to cling to the old, laborious method of taking the derivative, even when I was willing to tell them about my method.

With the previous definition, we have to maximize $x(250-x)$. Since $\sqrt{x(250-x)}\le \frac{x+250-x}{2}=125$, we have $x(250-x)\le 15625$; the maximum is $15625$.

This may not look much better than using derivatives. But soon the problems started becoming more annoying (the derivatives became complicated polynomials that were hard to solve, unlike that linear function in the example), but with a bit of thinking, I was able to adapt my method to deal with them successfully, and it became vastly quicker and simpler than defining a function and taking a derivative.

Of course, I would never have used my method if it worked more slowly for me than the official method. When I solve problems, I just use what's most useful, not what's the status quo.

Disclaimer: this does not mean I want everyone to use AM-GM on such optimization problems from now on. It worked for me because I already had extensive experience using it to apply and adapt it. But my point is that you don't have to use an official method if you know a way that is clearly superior.

BE DIFFERENT