Not true!

Let $a,b,c$ be real numbers such that $a+b+c=0$ and $a^2+b^2+c^2=1.$ Prove that $a^2b^2c^2\leq \frac{1}{54}.$ When does this equality hold true?

#Algebra

Easy Math Editor

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Comments

Without losing generality we assume that $ab\geq 0$ . From the 1st condition we have $a+b=-c$ replace this into the 2nd condition $a^2+b^2+(a+b)^2=1$ By C-S $a^2+b^2\geq\frac{(a+b)^2}{2}$ , so $1\geq\frac{3}{2}(a+b)^2 \Rightarrow c^2\leq\frac{2}{3}$ . Now we rewrite the 2nd condition $2(a^2+b^2)+2ab=1$ We have this inequality for all reals $a^2+b^2\geq 2ab$ , therefore $6ab\leq 1\Rightarrow a^2b^2\leq\frac{1}{36}$ $\therefore a^2b^2c^2\leq\frac{1}{54}$ The equality holds when $(a,b,c)=\bigg(\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{-2}{\sqrt{6}}\bigg);\bigg(\frac{-1}{\sqrt{6}},\frac{-1}{\sqrt{6}},\frac{2}{\sqrt{6}}\bigg)$

P C - 5 years ago

Let $\alpha=abc$ .

Now, as $0=(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)\\\implies 0=1+2(ab+bc+ca)\\\implies ab+bc+ca=-\frac{1}{2}$

Therefore, $a, b, c$ are the (real) roots of the following function:

$f(x)=x^3-\frac{1}{2}x-\alpha$

For the existence of three real roots, we must have: $f(x_1)f(x_2) \leq 0$ where $x_1$ and $x_2$ are the points of extrema of the function $f(x)$ .

Note that $f'(x_1)=f'(x_2)=0$ .

Since $f'(x)$ is a quadratic polynamial, it can be solved.

Solving, plugging it into the above inequality, we get: $|\alpha| \leq \frac{1}{3\sqrt6} \\\implies \alpha^2 \leq \frac{1}{54}$

With equality holding if and only if two of the real numbers $a, b, c$ are equal.

A Former Brilliant Member - 5 years ago

Note: It's slightly more straightforward to just take the cubic discriminant condition for 3 real roots.

Calvin Lin Staff - 5 years ago

The problem was I didn't remeber the cubic discriminant formula then. So, I used calculus in the solution.

A Former Brilliant Member - 5 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$