Find the range of values for abc and a + b + c

[anemone]

Let $a, b, c$ be positive real numbers satisfying \(\displaystyle \frac{1}{3}\le ab+bc+ca \le 3\).

Determine the range of values for

i) $abc$,

ii) $a+b+c$.

 

[Opalg]

Anemone and kaliprasad have noticed that nobody ever replied to this challenge problem. Here is my attempt, using MarkFL's favourite method of Lagrange multipliers.

To find the extreme points of $abc$ subject to the restraint $bc+ca+ab = k$ (where $\frac13\leqslant k\leqslant 3$), put the partial derivatives of $abc - \lambda(bc+ca+ab - k)$ (with respect to $a$, $b$ and $c$) equal to $0$: $$bc - \lambda (b+c) = 0,\qquad ca - \lambda (c+a) = 0,\qquad ab - \lambda (a+b) = 0.$$ Write those equations as \(\displaystyle \frac1\lambda = \frac1b + \frac1c = \frac1c + \frac1a = \frac1a + \frac1b\) to see that $a=b=c$. That is the unique extremal point of $abc$. It must be a maximum because if we take $b=c=\varepsilon$ and $a = \dfrac{k-\varepsilon^2}{2\varepsilon}$ then $bc+ca+ab = k$ but $abc = \frac12\varepsilon(k-\varepsilon^2) \to0$ as $\varepsilon\to0$. So $abc\to0$ towards the boundary of the set $\{(a,b,c)\in \mathbb{R}^3:a>0,\,b>0,\,c>0\}.$ Thus the maximum possible value of $abc$ occurs when $k=3$ and $a=b=c= abc =1$. The range of values of $abc$ is therefore the half-open interval $(0,1]$.

An exactly similar calculation for the sum $a+b+c$ shows that it can take arbitrarily large values (when $b=c= \varepsilon$, $a = \dfrac{k-\varepsilon^2}{2\varepsilon}$ and $\varepsilon\to0$). There is again a unique extremal point when $a=b=c$, but this time it is a minimum, occurring when $a=b=c=\frac13$ and $a+b+c=1$. So the range of values of $a+b+c$ is the interval $[1,\infty).$

 

[MarkFL]

Spoiler

I think the reason I favor the method of Lagrange multipliers is because I am ignorant of the AM-GM method. (Wink) anemone has tried to teach me this, but I have been quite slow on the uptake. (Giggle)

 

[anemone]

Spoiler

I think the reason I favor the method of Lagrange multipliers is because I am ignorant of the AM-GM method. (Wink) anemone has tried to teach me this, but I have been quite slow on the uptake. (Giggle)

Spoiler

Hahaha...that isn't the case, Mark! That is because you don't like AM-GM for some reason, the same reason I have zero interest with the LM, I guess...:p

 

[Opalg]

Spoiler

I think the reason I favor the method of Lagrange multipliers is because I am ignorant of the AM-GM method. (Wink) anemone has tried to teach me this, but I have been quite slow on the uptake. (Giggle)

[sp]Mention of the AM-GM method makes me see that this is the best way to approach this problem.View attachment 2344

In fact, $\frac13(bc + ca + ab) \geqslant \sqrt[3]{a^2b^2c^2}.$ So if $bc+ca+ab \leqslant3$ it follows that $(abc)^{2 / 3} \leqslant1$ and so $abc\leqslant 1$.

For the other part of the problem, add the inequalities $b^2 + c^2 \geqslant 2bc$, $c^2+a^2 \geqslant 2ca$ and $a^2+b^2 \geqslant 2ab$ to get $2(a^2+b^2+c^2) \geqslant 2(bc+ca+ab)$ and hence $a^2+b^2+c^2 \geqslant bc+ca+ab.$ It follows that $(a+b+c)^2 = a^2+b^2+c^2 + 2(bc+ca+ab) \geqslant 3(bc+ca+ab) \geqslant1.$ Therefore $a+b+c\geqslant1.$[/sp]