{"id":653,"date":"2025-10-09T15:28:15","date_gmt":"2025-10-09T15:28:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=653"},"modified":"2026-04-13T17:06:05","modified_gmt":"2026-04-13T17:06:05","slug":"critical-points-and-local-extrema-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/critical-points-and-local-extrema-apply-it\/","title":{"raw":"Critical Points and Local Extrema: Apply It","rendered":"Critical Points and Local Extrema: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find critical points using partial derivatives<\/li>\r\n \t<li>Apply the second derivative test<\/li>\r\n \t<li>Find absolute extrema on closed regions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Optimizing Solar Panel Efficiency<\/h2>\r\nYou work for a renewable energy company that installs solar panels on flat commercial rooftops. The energy output efficiency [latex]E[\/latex] at any position on a rooftop depends on two factors: [latex]x[\/latex], the distance in meters from the west edge, and [latex]y[\/latex], the distance in meters from the south edge. Factors like nearby buildings, ventilation units, and rooftop geometry create variations in sunlight exposure across the surface. Your job is to analyze efficiency models to find optimal panel placement. By locating critical points and classifying them, you can identify positions of maximum efficiency, minimum efficiency, or unstable transition zones. When working within a specific installation zone, you must also check boundary constraints to find the absolute best and worst positions.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322651[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322653[\/ohm_question]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322658[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322659[\/ohm_question]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find critical points using partial derivatives<\/li>\n<li>Apply the second derivative test<\/li>\n<li>Find absolute extrema on closed regions<\/li>\n<\/ul>\n<\/section>\n<h2>Optimizing Solar Panel Efficiency<\/h2>\n<p>You work for a renewable energy company that installs solar panels on flat commercial rooftops. The energy output efficiency [latex]E[\/latex] at any position on a rooftop depends on two factors: [latex]x[\/latex], the distance in meters from the west edge, and [latex]y[\/latex], the distance in meters from the south edge. Factors like nearby buildings, ventilation units, and rooftop geometry create variations in sunlight exposure across the surface. Your job is to analyze efficiency models to find optimal panel placement. By locating critical points and classifying them, you can identify positions of maximum efficiency, minimum efficiency, or unstable transition zones. When working within a specific installation zone, you must also check boundary constraints to find the absolute best and worst positions.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322651\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322651&theme=lumen&iframe_resize_id=ohm322651&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322653\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322653&theme=lumen&iframe_resize_id=ohm322653&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322658\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322658&theme=lumen&iframe_resize_id=ohm322658&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322659\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322659&theme=lumen&iframe_resize_id=ohm322659&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-653","chapter","type-chapter","status-publish","hentry"],"part":430,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/653","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/653\/revisions"}],"predecessor-version":[{"id":2857,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/653\/revisions\/2857"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/430"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/653\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=653"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=653"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=653"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=653"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}