Triangle inequality and max-min « Mathematics Hothouse

Triangle inequality and max-min

Here is a problem I culled from Prof. Titu Andreescu’s literature (Geometric maxima and minima) for IITJEE Mathematics:

Problem:

Find the greatest real number k such that for any triple of positive numbers a, b, c such that $kabc > a^{3}+b^{3}+c^{3}$ , there exists a triangle with side lengths a, b, c.

Solution:

We have to find the greatest real number k such that for any a, b, $c > 0$ with $a + b \leq c$ , we have $kabc \leq a^{3}+b^{3}+c^{3}$ . First take $b=a$ and $c=2a$ . Then, $2ka^{3} \leq 10a^{3}$ , that is, $k \leq 5$ . Conversely, let $k=5$ . Set $c=a+b+x$ , where $x \geq 0$ . Then,

$a^{3}+b^{3}+c^{3}-5abc =2(a+b)(a-b)^{2}+(ab+3a^{2}+3b^{2})x+3(a+b)x^{2}+x^{3} \geq 0$ .

QED.

More later,

Nalin Pithwa

This entry was written by Nalin Pithwa, posted on May 19, 2016 at 7:35 am, filed under IITJEE Advanced Mathematics, IITJEE Mains and tagged maxima minima, triangle inequality. Bookmark the permalink. Follow any comments here with the RSS feed for this post. Post a comment or leave a trackback: Trackback URL.

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