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""mcs“ —2015/5/18 —1:43 — page 12 —#20

12 Chapter 1 What is a Proo.7

Example

Theorem 1.5.1. If $\mathrm{0}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{x}\underset{\mathrm{‾}}{\mathrm{<}}\mathrm{2}\mathrm{,}$ $\mathit{t}\mathit{h}\mathit{e}\mathit{n}\mathrm{-}{\mathit{x}}^{\mathrm{3}}\mathrm{+}\mathrm{4}\mathit{x}\mathrm{+}\mathrm{1}\mathrm{>}\mathrm{0}\mathrm{.}$

Before we write a proof of this theorem, we have to do some scratchwork to figure out why it is true.

The inequality certainly holds for $\mathit{x}\mathrm{=}\mathrm{0}$; then the left side is equal to 1 and $\mathrm{1}\mathrm{>}\mathrm{0}$. As $\mathit{x}$ grows, the $\mathrm{4}\mathit{x}$ term (which is positive) initially seems to have greater magnitude than $\mathrm{-}{\mathit{x}}^{\mathrm{3}}$ (which is negative). For example, when $\mathit{x}\mathrm{=}\mathrm{1}$, we have $\mathrm{4}\mathit{x}\mathrm{=}\mathrm{4}$, but $\mathrm{-}{\mathit{x}}^{\mathrm{3}}\mathrm{=}\mathrm{-}\mathrm{1}$ only. In fact, it looks like $\mathrm{-}{\mathit{x}}^{\mathrm{3}}$ doesn't begin to dominate until $\mathit{x}\mathrm{>}\mathrm{2}$. So it seems the $\mathrm{-}{\mathit{x}}^{\mathrm{3}}\mathrm{+}\mathrm{4}\mathit{x}$ part should be nonnegative for all $\mathit{x}$ between $\mathrm{0}$ and 2, which would imply that $\mathrm{-}{\mathit{x}}^{\mathrm{3}}\mathrm{+}\mathrm{4}\mathit{x}\mathrm{+}\mathrm{1}$ is positive.

So far, so good. But we still have to replace all those ""seems like“ phrases with solid, logical arguments. We can get a better handle on the critical $\mathrm{-}{\mathit{x}}^{\mathrm{3}}\mathrm{+}\mathrm{4}\mathit{x}$ part by factoring it, which is not too hard:

$\mathrm{-}{\mathit{x}}^{\mathrm{3}}\mathrm{+}\mathrm{4}\mathit{x}\mathrm{=}\mathit{x}\mathrm{\left(}\mathrm{2}\mathrm{-}\mathit{x}\mathrm{\right)}\mathrm{\left(}\mathrm{2}\mathrm{+}\mathit{x}\mathrm{\right)}$

Aha! For $\mathit{x}$ between $\mathrm{0}$ and 2, all of the terms on the right side are nonnegative. And a product of nonnegative terms is also nonnegative. Let's organize this blizzard of observations into a clean proof.

Proo. Assume $\mathrm{0}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{x}\underset{\mathrm{‾}}{\mathrm{<}}\mathrm{2}$. Then $\mathit{x}\mathrm{,}$ $\mathrm{2}\mathrm{-}\mathit{x}$, and $\mathrm{2}\mathrm{+}\mathit{x}$ are all nonnegative. Therefore, the product of these terms is also nonnegative. Adding 1 to this product gives a positive number, so:

$\mathit{x}\mathrm{\left(}\mathrm{2}\mathrm{-}\mathit{x}\mathrm{\right)}\mathrm{\left(}\mathrm{2}\mathrm{+}\mathit{x}\mathrm{\right)}\mathrm{+}\mathrm{1}\mathrm{>}\mathrm{0}$

Multiplying out on the left side proves that

$\mathrm{-}{\mathit{x}}^{\mathrm{3}}\mathrm{+}\mathrm{4}\mathit{x}\mathrm{+}\mathrm{1}\mathrm{>}\mathrm{0}$

as claimed. $\mathrm{\blacksquare }$

There are a couple points here that apply to all proofs:

$\mathrm{•}$ You'll often need to do some scratchwork while you're trying to figure out the logical steps of a proof. Your scratchwork can be as disorganized as you like—full of dead‐ends, strange diagrams, obscene words, whatever. But keep your scratchwork separate from your final proof, which should be clear and concise.

$\mathrm{•}$ Proofs typically begin with the word ""Proof“ and end with some sort of de‐ limiter like $\mathrm{\square }$ or ""QED.“ The only purpose for these conventions is to clarify where proofs begin and end.