{"id":34781,"date":"2025-06-23T19:31:14","date_gmt":"2025-06-23T19:31:14","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=34781"},"modified":"2025-06-23T19:31:16","modified_gmt":"2025-06-23T19:31:16","slug":"concavity-assign-38-i-j-70-23-graph-of-f-the-graph-of-f-the-derivative-of-the-function-f-is-shown-above","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/concavity-assign-38-i-j-70-23-graph-of-f-the-graph-of-f-the-derivative-of-the-function-f-is-shown-above\/","title":{"rendered":"Concavity Assign 38 I J 70 23 Graph of f&#8217; The graph of f&#8217;, the derivative of the function f, is shown above."},"content":{"rendered":"\n<p>Concavity Assign 38 I J 70 23 Graph of f&#8217; The graph of f&#8217;, the derivative of the function f, is shown above. On which of the following open intervals is the graph of f concave down? (-5, -3) and (1, 6) (-3, 1) and (6, 8) (-1, 4) (4, 8)<br>Unit 5 Progress Check: MCQ Part B Q6: Representations of Functions &#8211; Concavity Assign 38 I J 70 23 Graph of f&#8217; The graph of f&#8217;, the derivative of the function f, is shown above. On which of the following open intervals is the graph of f concave down? (-5, -3) and (1, 6) (-3, 1) and (6, 8) (-1, 4) (4, 8)<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"957\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-264.png\" alt=\"\" class=\"wp-image-34803\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-264.png 957w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-264-280x300.png 280w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-264-768x822.png 768w\" sizes=\"auto, (max-width: 957px) 100vw, 957px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The correct answer is: <strong>(\u20135, \u20133) and (1, 6)<\/strong>.<\/p>\n\n\n\n<p>To determine where the function <em>f<\/em> is concave down, we analyze its second derivative <em>f\u2033(x)<\/em>. Since we&#8217;re given the graph of <em>f\u2032(x)<\/em> (the first derivative), we must identify where <em>f\u2032(x)<\/em> is decreasing. When <em>f\u2032(x)<\/em> is decreasing, its slope is negative, which implies that <em>f\u2033(x) &lt; 0<\/em>, and thus <em>f<\/em> is concave down.<\/p>\n\n\n\n<p>From the graph:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>On the interval <strong>(\u20135, \u20133)<\/strong>, the graph of <em>f\u2032(x)<\/em> clearly decreases from a higher positive value to a lower one. This indicates that the slope of <em>f<\/em> is decreasing, so <em>f\u2033(x)<\/em> is negative. Therefore, <em>f<\/em> is concave down on this interval.<\/li>\n\n\n\n<li>On the interval <strong>(1, 6)<\/strong>, the graph again shows a decreasing trend \u2014 <em>f\u2032(x)<\/em> declines from a positive value, approaches zero, and may even become negative. This confirms that <em>f\u2033(x)<\/em> remains negative over this interval, so <em>f<\/em> is concave down here as well.<\/li>\n<\/ul>\n\n\n\n<p>Now consider the other choices:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>(\u20133, 1)<\/strong> contains a portion where <em>f\u2032(x)<\/em> is increasing, meaning <em>f\u2033(x)<\/em> is positive for at least part of it. So, <em>f<\/em> is not concave down throughout.<\/li>\n\n\n\n<li><strong>(6, 8)<\/strong> shows a decreasing portion of <em>f\u2032(x)<\/em>, so <em>f<\/em> may be concave down there, but the interval alone is not paired with another correct one like in option A.<\/li>\n\n\n\n<li><strong>(\u20131, 4)<\/strong> includes part of an increasing slope, which means <em>f\u2033(x)<\/em> is positive there.<\/li>\n\n\n\n<li><strong>(4, 8)<\/strong> also contains parts where the slope of <em>f\u2032(x)<\/em> increases and then decreases, so the concavity isn&#8217;t consistently downward.<\/li>\n<\/ul>\n\n\n\n<p>Thus, <strong>only the intervals (\u20135, \u20133) and (1, 6)<\/strong> reflect portions where <em>f\u2032(x)<\/em> is strictly decreasing throughout, confirming the graph of <em>f<\/em> is concave down there.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-594.jpeg\" alt=\"\" class=\"wp-image-34814\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-594.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-594-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-594-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Concavity Assign 38 I J 70 23 Graph of f&#8217; The graph of f&#8217;, the derivative of the function f, is shown above. On which of the following open intervals is the graph of f concave down? (-5, -3) and (1, 6) (-3, 1) and (6, 8) (-1, 4) (4, 8)Unit 5 Progress Check: MCQ [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-34781","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34781","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=34781"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34781\/revisions"}],"predecessor-version":[{"id":34816,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/34781\/revisions\/34816"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=34781"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=34781"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=34781"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}