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I am self learning math proofs and I'm just having trouble understanding something. Say that I was to prove that any positive function is greater than $0$. Take the random example $x^2 +5x+ 7 > 0$ for all real numbers x or smt.

I don't know how the common methods like proof by contrapositive, contradiction, etc. applies here.

I was originally planning to show that there exists no real roots... but that does not show that the equation is positive/greater than $0$.

(No need to necessarily go through with that example, I just sorta want to know the general mindset when going through questions like this.)

Gary Liang
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Learningmath
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    For this you could try completing the square. – Sean Roberson May 17 '23 at 01:27
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    You were "to prove that any positive function is greater than $0$"? How do you define "positive function"? – peterwhy May 17 '23 at 01:31
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    As a student, I always had the impression that coming up with a proof was more art than science, or at least needs certain amount of guesswork. Proving inequalities is often done via transforming it into another known inequality, or coming up with some wierd, looser but strict enough bounds for certain variables. I think for these problems, reading many proofs and taking note of the techniques used is the most helpful. – 이희원 May 17 '23 at 02:16

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In general, there are many ways of proving inequalities and recognising what techniques to use in different situations takes practice. For a quadratic function like $f(x) = x^2 +5x+7$, here are just some ways:

  • Completing the square: You could rewrite $f(x) = (x+5/2)^2 + 3/4$, and since square numbers are non-negative, you could say that $f(x) >0$ for all real $x$.
  • Drawing a graph: Drawing a graph of a quadratic function can help you understand why it is always positive. The graph will be a "upwards" parabola since the coefficient of $x^2$ is positive, so if you can find the vertex then that would be sufficient. (Note: some people may not consider this a proof but it is still a good way to gain intuition on why the function is always positive)
  • Finding the stationary points: If you know calculus, you could differentiate $f(x)$ and find out where the stationary point is, and then show that it is a global minimum and that it lies above the $x$-axis.
  • Showing there are no real roots: Your idea here is actually quite good. For example, you could find the discriminant $\Delta = b^2 - 4ac = -3$, which shows there are no real roots. Since it's a quadratic function, you now know that it is either always positive or always negative. Now you know that, say $f(0) = 7 > 0$, which means $f(x) > 0$ for all real $x$.
Gary Liang
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  • Great, Thank you! I have been trying too hard to stick to formats in textbooks that I forgot that I can go about many ways algebraically as direct proof. – Learningmath May 17 '23 at 02:20