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I was given "If $f'(x)>0$ for every $x\in(a, b),$ then $f$ is increasing on $[a, b].$" and I need to write it out symbolically. I have gotten $$∀x\in(a,b) ~(f'(x)>0)\to (x<y\to f(x)<f(y))$$

I don't know how to rewrite $f'(x)>0$ in more basic terms like I did with increasing. Can anybody help me with this?

Graham Kemp
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George
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  • This is incorrect; you are saying instead that for each $x$, if the derivative of $f$ at $x$ is positive, then.... something because $y$ is free in your statement. It doesn't say what you need it to say. – Arturo Magidin Oct 03 '18 at 04:05

2 Answers2

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You need to qualify $x$ and $y$ in the 'then' part. Otherwise, you're nearly there. Here's what I would write: $$(x\in(a,b) \implies f'(x)>0)\implies[((x\in[a,b])\land(y\in[a,b])\land (x<y))\implies f(x)\le f(y)].$$

Adrian Keister
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It looks almost correct to me, save that you need to quantify the free variables on the RHS.$$∀x\in(a,b) ~(f'(x)>0)\to \boxed{????}(x<y\to f(x)<f(y))$$

Graham Kemp
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  • Well, your edit introduced unbalanced parentheses, so it's hard to tell. Also, you potentially changed the meaning, depending on where the parentheses was supposed to be.... The original statement seemed to be a single quantifier on an implication, whereas your edit has the quantifier in the antecedent. That required the OP to fix, not you to guess... – Arturo Magidin Oct 03 '18 at 04:07