{"id":11779,"date":"2021-07-06T11:35:01","date_gmt":"2021-07-06T11:35:01","guid":{"rendered":"http:\/\/www.max-sperling.bplaced.net\/?p=11779"},"modified":"2021-07-06T11:35:01","modified_gmt":"2021-07-06T11:35:01","slug":"math-exercise-8-analysis","status":"publish","type":"post","link":"http:\/\/www.max-sperling.bplaced.net\/?p=11779","title":{"rendered":"Math \u2013 Exercise 8 (Analysis)"},"content":{"rendered":"<p><strong>Given<\/strong><br \/>\n<center>\\(f(x,y) = x^2 + y^2\\)<\/center><\/p>\n<hr>\n<p><strong>Find<\/strong><br \/>\n<center>All extremum of that function and their types.<\/center><\/p>\n<hr>\n<p><strong>Solution<\/strong><br \/>\n<center><br \/>\n\\(<br \/>\n\\begin{array}{l}<br \/>\n  \\text{Extremum:} \\\\<br \/>\n  \\nabla f(x,y) = (\\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}) = (2x, 2y) = 0 \\\\\\\\<br \/>\n  \\begin{eqnarray} \\text{I:}\\ &#038; 2x = 0 \\\\ \\text{II:}\\ &#038; 2y = 0 \\end{eqnarray} \\\\\\\\<br \/>\n  E_1 = (0,0) \\\\\\\\\\\\<br \/>\n  \\text{Types:} \\\\<br \/>\n  H_f(x,y) = \\begin{pmatrix} \\frac{\\partial^2 f}{\\partial x^2} &#038; \\frac{\\partial^2 f}{\\partial x \\partial y} \\\\ \\frac{\\partial^2 f}{\\partial y \\partial x} &#038; \\frac{\\partial^2 f}{\\partial y^2} \\end{pmatrix} = \\begin{pmatrix} 2 &#038; 0 \\\\ 0 &#038; 2 \\end{pmatrix} \\\\\\\\<br \/>\n  H_f(0,0) = \\begin{pmatrix} 2 &#038; 0 \\\\ 0 &#038; 2 \\end{pmatrix} \\\\<br \/>\n  H_1(0,0) = 2 \\gt 0\\\\<br \/>\n  H_2(0,0) = \\begin{vmatrix} 2 &#038; 0 \\\\ 0 &#038; 2 \\end{vmatrix} = 2 \\cdot 2 &#8211; 0 \\cdot 0 = 4 \\gt 0 \\\\\\\\<br \/>\n  H_f(0,0)\\ \\text{is positive-define and therefore}\\ E_1\\ \\text{is a minima.}<br \/>\n\\end{array}<br \/>\n\\)<br \/>\n<\/center><\/p>\n<hr>\n<p><strong>Answer<\/strong><br \/>\n<center>\\(E_1 = (0,0),\\ \\text{minima}\\)<\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given Find All extremum of that function and their types. Solution Answer<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[47],"tags":[],"_links":{"self":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts\/11779"}],"collection":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11779"}],"version-history":[{"count":0,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=\/wp\/v2\/posts\/11779\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11779"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11779"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.max-sperling.bplaced.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11779"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}