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Proposition: Suppose that $a$, $b$, and $c$ are real numbers with $a \not = 0$. Prove that, if $x^∗ = \dfrac{−b}{2a}$ is a maximizer of the function $f(x) = ax^2 + bx + c$, then a < 0.

A (hypothesis): $x^∗ = \dfrac{−b}{2a}$ is a maximizer of the function $f(x) = ax^2 + bx + c$ where $a \not = 0$, $b$, and $c$ are real numbers.

B (conclusion): $a < 0$

A1: For all real numbers $x$, $x^* = \dfrac{−b}{2a}$ is a maximiser of the function $f(x) = ax^2 + bx + c$.

A1 rephrases A using the universal quantifier "for all".

A2: Let $x \in \mathbf{R}$ and $x = \dfrac{-b}{2a}$.

A3: $f(x) = ax^2 + bx + c$

$\implies f\left(\dfrac{−b}{2a}\right) = a\left(\dfrac{−b}{2a}\right)^2 + b\left(\dfrac{−b}{2a}\right) + c$

$ = \dfrac{ab^2}{4a^2} - \dfrac{b^2}{2a} + c$

$ = \dfrac{b^2}{4a} - \dfrac{b^2}{2a} + c$

$ = \dfrac{b^2 - 2b^2}{4a} + c$

$ = \dfrac{-b^2}{4a} + c$

A4: $\dfrac{-b^2}{4a} + c \ge ax^2 + bx + c$

$\implies \dfrac{-b^2}{4a} \ge ax^2 + bx$

$\therefore a < 0$

$Q.E.D.$

I would greatly appreciate it if people could please take the time to look over my proof and provide feedback.

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    If for all $x\in\mathbb R$ we have $ax^2+bx+c\leq-\frac{b^2}{4a}+c$ so indeed, $a<0$. I think, it's true. – 2017-02-03
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    @MichaelRozenberg So you don't see any errors in my proof? – 2017-02-03
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    I think you need to add that for $a>0$ the maximum does not exist. I don't see errors. – 2017-02-03
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    @MichaelRozenberg I see what you mean. Thank you for the assistance. :) – 2017-02-03

2 Answers 2

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$$ax^2+bx+c=a\left(x^2+\frac{b}{a}x+\frac{c}{a}\right)=a\left(\left(x+\frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a^2}\right)\leq-\frac{b^2-4ac}{4a}$$ The equality occurs for $x=-\frac{b}{2a}$.

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    Thanks for the response. Are you saying that my proof is incorrect? If so, which specific part? – 2017-02-03
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In your (A4) part, there is a big jump to reach the “therefore” part.

Following the post by @MichaelRozenberg, we have $ax^2+bx+c= … = a\left(\left(x+\frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a^2}\right)$

$= a\left(x+\frac{b}{2a}\right)^2 -\frac{b^2-4ac}{4a}$

$\leq-\frac{b^2-4ac}{4a}$ for all x ONLY when $a\left(x+\frac{b}{2a}\right)^2$ is negative

This further means 'a' must be negative since $\left(x+\frac{b}{2a}\right)^2$ is always positive for all x.

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    I don't understand why this is necessary? Doesn't A4 already show that $a < 0$? It seems like this involves rearranging things for no reason? – 2017-02-03
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    What reason supports your conclusion of a < 0? Has that been explained? – 2017-02-03
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    $\dfrac{-b^2}{4a} \ge ax^2 + bx$ since $x = \dfrac{-b}{2a}$ is a maximiser of $f(x) = ax^2 + bx + c$. And we can see, algebraically, that the only way for $\dfrac{-b^2}{4a} \ge ax^2 + bx$ is if $a < 0$. So using the assumed information from A, we must necessarily have $a < 0$. – 2017-02-03
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    @ThePointer I don’t see why we must have a < 0 (algebraically) from $\dfrac {−b^2}{4a} \ge ax^2+bx$. – 2017-02-03