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I'm taking my first math course that requires me to write proofs, and even though I understand most of the course material, I'm struggling with actually proving things in a rigorous way.

For example, we're asked to show (using what they've called the trivial method of proof) that $$\text{if } x^3-5x-1\geq 0\text{,} \; \text{then}\; (x-1)(x-3)\geq -2\text{.}$$
I get that the trivial method means were dont really need to use the "if" case to show that the "then" case is true, and I can see that it is true, but how do I formally show it?

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Lincoln77
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    That's a really weird thing to call a method of proof, and I think it's a bad idea. Regardless: Forget about the "if" part and notice that $(x - 1)(x - 3)$ is a quadratic that's minimized at $x = 2$. –  Mar 29 '16 at 05:37
  • Hey thanks for your help, this may sound really dumb, but how can you see that its minimised at 2? – Lincoln77 Mar 29 '16 at 05:41
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    Draw the graph and notice that $x = 2$ is half way between the roots. Or do a line of calculus. (Or expand and complete the square). –  Mar 29 '16 at 05:42

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HINT: You can rewrite $(x-1)(x-3)\geq -2$ as $(x -2)^2 + 1 \geq 0$

  • hey, thanks for your answer, I understand the math but Im just after some help with the formal aspect of writing the proof. – Lincoln77 Mar 29 '16 at 05:53
  • I think something is wrong with the sentence: if .. then, because what you have after then is true for all x, regardless the condition from "if". –  Mar 29 '16 at 05:56
  • yep, thats why its asking for a 'trivial' proof, my question was how to formally show that (x−1)(x−3)≥−2 is always true. As in, not so much the math side of it, but the proof writing side. – Lincoln77 Mar 29 '16 at 05:58
  • I just have giving you a hint! –  Mar 29 '16 at 05:59
  • yes, thanks for that, but im not really looking for a hint so much as advice on how to formally structure and write the proof. how to state on paper, the problem, and get to QED, in words. – Lincoln77 Mar 29 '16 at 06:05
  • The difficulty here is that the problem isn't one which needs rigorous formal language to prove it. This is compounded by the fact that we don't know the standards/levels of formality/rigor expected for your class. – Ian Miller Mar 29 '16 at 06:22
  • hey ian, yeah its annoying, the class is putting a LOT of stress on showing things with watertight logical proof, so, while i understand the hint, im just not sure if i can use it as the graphs of the two functions (x−1)(x−3) and (x−2)^2+1 are different? i agree that the simplicity of the question add a degree of difficulty to it that is ironic, but yeah, i really have no idea how to even start 'proving' this. – Lincoln77 Mar 29 '16 at 07:39
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You can structure your proof as follows: $$(x-1)(x-3)\ge-2$$ $$x^2-4x+5\ge0$$ $$(x-2)^2+1\ge0$$ $$\text{True, as }u^2\ge0 \forall u\in\mathbb R\text{, which follows that }u^2+1\ge0.$$

Will Sherwood
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