Is my method for proving this implication correct?

Question

My professor put up the lecture slides from today's lecture and I decided to go over the proof again since I didn't quite catch it in class. I think I found something wrong with her proof. Please let me know if mine is correct or if I am wrong:

Proving an Implication:

Theorem: If $0\le x\le 2$ then $-x^3+4x+1>0$

Proof:

Assume $0\le x\le 2$
Then $0\le x^2\le 4$ (since $0\le a,b\quad \& \quad a\le b\Rightarrow a^2\le b^2$)
$-4\le -x^2\le 0$
$0\le -x^2+4\le 4$
$0\le x(-x^2+4)\le 4x$
$0\le -x^3+4x\le 4x$
$1\le -x^3+4x+1\le 4x+1$
More specifically, $-x^3+4x+1>0$

The proof from the lecture slides:

Your proof is fine - which does not imply her proof is wrong, though — Hagen von Eitzen, Oct 18 '15 at 20:21
@HagenvonEitzen Please look over the proof from the lecture. I added it. — Cherry_Developer, Oct 18 '15 at 20:27
Looks good. But that doesn't mean there's anything wrong with hers. Mine would be far shoter. (O<x < 2 so x^2 < 4x so x^3 < 4x so -x^3 - 4x + 1 > 1 >0) — fleablood, Oct 18 '15 at 20:28
The minus in front of $4x$ in the slides has to be a typo. Just as in @fleablood's version. — String, Oct 18 '15 at 20:29
@fleablood As string said, the minus signs are slightly off. — Cherry_Developer, Oct 18 '15 at 20:31
In her proof lines 4-8 have a typo $-4$ must be $+4$. In the theorem statement it is correct sign. — A.Γ., Oct 18 '15 at 20:31
In fact what the heck is different between yours and hers. They're same so far as I can tell (except you stated but never used <= 4x+1) — fleablood, Oct 18 '15 at 20:32
BTW, I would not hesitate to jump directly to $$-x^3+4x=x(4-x^2)\geq 0$$ from which the result should immediately be clear, but that is just a matter of taste. — String, Oct 18 '15 at 20:35
@String Would you be able to state your first answer in the comments and the reason why you wouldn't immediately jump to that conclusion in an answer below? I will accept it and +1. — Cherry_Developer, Oct 18 '15 at 20:41
@Cherry_Developer: I tried, but you are not obliged to accept or upvote my answer. But feel free to comment, suggest improvements, ask for clarifications etc. — String, Oct 18 '15 at 21:18

String · Accepted Answer · 2015-10-18T21:07:03.697

This is merely a matter of taste, but I would not hesitate to jump directly to $$ -x^3+4x=x(4-x^2) $$ which must be greater than or equal to zero since both factors, $x$ and $4-x^2$, are evidently non-negative for $0\leq x\leq 2$. This is merely summarizing what you already stated yourself. So $-x^3+4x\geq 0$ and thus $$ -x^3+4x+1\geq 1>0 $$

Regarding the slides, it does NOT make sense to state that $$ 0\leq -x^3-4x=-(x^3+4x) $$ which is clearly false for any positive value of $x$. Thus the minus sign in $-4x$ must be a typo in the slides.

Is my method for proving this implication correct?

1 Answers1