{"id":303,"date":"2019-10-11T12:00:44","date_gmt":"2019-10-11T17:00:44","guid":{"rendered":"http:\/\/blogs.acu.edu\/mu_sigma\/?p=303"},"modified":"2019-11-01T11:59:34","modified_gmt":"2019-11-01T16:59:34","slug":"problem-6-fall-2019-two-handy-inequalities","status":"publish","type":"post","link":"https:\/\/blogs.acu.edu\/mu_sigma\/2019\/10\/11\/problem-6-fall-2019-two-handy-inequalities\/","title":{"rendered":"Problem 6 – Fall 2019 – Two Handy Inequalities"},"content":{"rendered":"

<\/h3>\n

Two Handy Inequalities<\/b><\/span><\/p>\n

This problem was suggested by Alexander Karabegov.<\/i><\/span><\/p>\n

Prove the following inequalities are true for all positive real numbers x<\/i> and y<\/i>, with equality holding if and only if x<\/i> = y<\/i>.<\/p>\n

<\/p>\n

Hint: (x<\/i>–y<\/i>)2<\/sup> is greater than or equal to 0.<\/p>\n

Please submit all work to mathpotw@acu.edu no later than 5:00 PM on Thursday, October 17.\u00a0<\/em><\/p>\n

There were no correct submissions for this problem.\u00a0<\/em><\/p>\n

Solution to Two Handy Inequalities<\/b><\/span><\/p>\n

In fact, we can prove a bit more than the problem asked. We will show that<\/p>\n

with either equality holding if, and only if, x<\/i> = y<\/i>.<\/p>\n

It is certainly true that (x<\/i>–y<\/i>)2<\/sup> > 0 when, and only when, x<\/i> is not equal to y<\/i>. From this expression we derive the inequality<\/p>\n

x<\/i>2<\/sup> + y<\/i>2<\/sup> > 2xy<\/i> (*)and from this inequality, we derive two more inequalities.<\/p>\n

First, by adding x<\/i>2<\/sup> + y<\/i>2<\/sup> to both sides of inequality (*), we find that 2(x<\/i>2<\/sup>+y<\/i>2<\/sup>) > x<\/i>2<\/sup> + 2xy<\/i> + y<\/i>2<\/sup>.<\/p>\n

Next, by adding 2xy<\/i> to both sides of the expression (*), we see that x<\/i>2<\/sup> + 2xy<\/i> + y<\/i>2<\/sup> > 4xy<\/i>.<\/p>\n

The claimed inequalities follow easily from the two expressions we just derived.<\/p>\n","protected":false},"excerpt":{"rendered":"

Two Handy Inequalities This problem was suggested by Alexander Karabegov. Prove the following inequalities are true for all positive real numbers x and y, with equality holding if and only if x = y. Hint: (x–y)2 is greater than or equal to 0. Please submit all work to mathpotw@acu.edu no later than 5:00 PM on […]<\/p>\n","protected":false},"author":130,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[181534],"tags":[],"class_list":["post-303","post","type-post","status-publish","format-standard","hentry","category-problem-of-the-week"],"_links":{"self":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/303","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/users\/130"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/comments?post=303"}],"version-history":[{"count":4,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/303\/revisions"}],"predecessor-version":[{"id":312,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/303\/revisions\/312"}],"wp:attachment":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/media?parent=303"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/categories?post=303"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/tags?post=303"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}