Question on proof for: If $a$, $b$, and $c$ are real numbers with $a<0$, then there is a real number $y$ such that for every real $x$, $ax^2+bx+c≤y$.

Question

I have a very weak background in math and am (slowly) attempting to learn more. I'm starting with "How to Read and Do Proofs" by Daniel Solow (5th edition). In Chapter 7 (pg.83), he presents this proposition:

If a, b, and c are real number with a < 0, then there is a real number y such that for every real x, $ax^2 + bx +c \le y$.

He begins the proof with:

Let $y = \frac{4ac-b^2}{4a}$.

This is most likely a naive question, but where is he getting the beginning of the proof from? Thanks, in advance.

score 2 · Accepted Answer · answered Jul 11 '23 at 20:38

Many proofs in books, papers and such have the great benefit of hindsight. Figuring out what to consider is easy once you have completed a proof!

In this case, note that $y$ is the $y$-coordinate of the turning point of the parabola. To see this, complete the square or differentiate. I would guess however, that this is actually what the remainder of the proof does.

So to figure out $y$ in the first place, you could have thought about what the maximum of $ax^2 + bx + c$ would be, used completing the square to find it, then constructed a proof afterwards using your found value of $y$ to make a rigorous argument.

Question on proof for: If $a$, $b$, and $c$ are real numbers with $a<0$, then there is a real number $y$ such that for every real $x$, $ax^2+bx+c≤y$.

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