[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-detail-522856-en":59,"doc-seo-522856-105":79},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":5,"data":60},{"doc_id":61,"user_id":62,"nickname":63,"user_avatar":64,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":66,"doc_content":67,"file_id":68,"file_url":69,"file_type":70,"file_size":71,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":72,"language_code":73,"site_id":74,"html_lang":73,"table_of_contents":75,"faqs":76,"seo_title":77,"seo_description":66,"update_tm":78,"read_time":52},522856,2336474466712,"Quinn Holloway","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Intervals of Concavity - Worksheet 04 - Kuta Software Infinite Calculus","Worksheet on intervals of concavity for multiple functions, requiring identification of x-coordinates of inflection points, determination of discontinuities, and classification of open intervals where each function is concave up versus concave down. Includes problems for polynomial and rational expressions, specifying inflection points or noting when none exist, and listing concavity intervals separated by critical x-values. Concludes with a critical thinking sketch problem enforcing conditions on f, derivatives, and sign changes of f'' across x = 1.","Kuta Software-Infinite Calculus Name   \nIntervals of Concavity Date  Period   \nFor each problem, find the x-coordinates of all points of inflection, find all discontinuities, and find the open intervals where the function is concave up and concave down.  \n1) y = x 3 − 3x2 + 4  \n2) y = x 3 − 2x2 − 2  \n3) y = x 4 + x 3 − 3x2 + 1  \n1  \n4) y =  \nx − 3  \nx 3  \n5) y = − ~~ ~~  \nx 2 − 4  \n2  \n6) y = (5x + 30) 3  \n4 1  \n7) y = − ~~ ~~316(x − 1) 3 − 32(x − 1) 3 + 2  \nCritical thinking question:  \n8) Sketch a continous curve y = f (x) where f ( 1) = 0, f ' (0) = 0, f ' (2) = 0, f '' (x) \u003C 0 for x \u003C 1, and f '' (x) > 0 for x > 1.  \n\n| | |  |  |  |  |  |  | y |  |  |  |  |  |  | |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n|  |  |  |  |  |  |  | 8 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 6 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 4 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 2 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n| −8 | − | 6 | − | 4 | − | 2 |  |  | 2 |  | 4 |  | 6 |  |  | 8 x |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −2 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −4 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −6 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  | −8\u003Cbr>|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n\nKuta Software-Infinite Calculus Name   \nIntervals of Concavity Date  Period   \nFor each problem, find the x-coordinates of all points of inflection, find all discontinuities, and find the open intervals where the function is concave up and concave down.  \n1) y = x 3 − 3x2 + 4  \nInflection point at: x = 1 No discontinuities exist.  \nConcave up: ( 1, ∞) Concave down: (−∞, 1)  \n2) y = x 3 − 2x2 − 2  \nInflection point at: x = 23 No discontinuities exist.  \nConcave up: (~~ ~~23~~ ~~ , ∞) Concave down: (− ∞, ~~ ~~23)  \n3) y = x 4 + x 3 − 3x2 + 1  \nInflection points at: x = −1, ~~ ~~12 No discontinuities exist.  \nConcave up: (−∞, −1),(~~ ~~12~~ ~~ , ∞) Concave down: (−1, ~~ ~~12)  \n4) y = x~~ ~~~~ ~~3  \nNo inflection points exist. Discontinuity at: x = 3  \nConcave up: (3, ∞) Concave down: (−∞, 3)  \nx 3  \n5) y = − ~~ ~~  \nx 2 − 4  \nInflection point at: x = 0 Discontinuities at: x = −2, 2  \nConcave up: (−∞, −2), (0, 2) Concave down: (−2, 0), (2, ∞)  \n2  \n6) y = (5x + 30) 3  \nNo inflection points exist. No discontinuities exist.  \nConcave up: No intervals exist. Concave down: (−∞, −6), (−6, ∞)  \n4 1  \n7) y = − ~~ ~~316(x − 1) 3 − 32(x − 1) 3 + 2  \nInflection points at: x = 1, 5 No discontinuities exist.  \nConcave up: ( 1, 5) Concave down: (−∞, 1), (5, ∞)  \nCritical thinking question:  \n8) Sketch a continous curve y = f (x) where f ( 1) = 0, f ' (0) = 0, f ' (2) = 0, f '' (x) \u003C 0 for x \u003C 1, and f '' (x) > 0 for x > 1.  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaitDsDUUrONu0","https://ap.wps.com/l/cbCaitDsDUUrONu0","pdf",37544,"English","en",105,"# Problems: Intervals of Concavity\n## Problem 1-7\n## Critical thinking question 8","[{\"question\":\"What does the worksheet ask you to find for each function?\",\"answer\":\"For each problem, find the x-coordinates of all inflection points, locate any discontinuities, and determine the open intervals where the function is concave up and concave down.\"},{\"question\":\"How are concavity intervals reported?\",\"answer\":\"The worksheet lists concave up intervals and concave down intervals as open intervals, written in forms such as (−∞, a), (a, b), or (a, ∞), based on where f'' changes sign.\"},{\"question\":\"What is included in the critical thinking question?\",\"answer\":\"The worksheet includes a sketching task for a continuous curve y = f(x) with given values of f(1), f'(0), f'(2), and specified inequality conditions on f''(x) for x \\u003c 1 and x \\u003e 1.\"}]","Intervals of Concavity - Worksheet 04 - Kuta Software Infinite Calculus | PDF",1791426238,{"code":4,"msg":80,"data":81},"ok",{"site_id":74,"language":73,"slug":82,"title":65,"keywords":83,"description":66,"schema_data":84,"social_meta":138,"head_meta":140,"extra_data":142,"updated_unix":143},"intervals-of-concavity-worksheet-04-kuta-software-infinite-calculus","",{"@graph":85,"@context":137},[86,100,120],{"@type":87,"itemListElement":88},"BreadcrumbList",[89,93,95,98],{"item":90,"name":91,"@type":92,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":94,"name":9,"@type":92,"position":14},"https://docshare.wps.com/document/",{"item":96,"name":20,"@type":92,"position":97},"https://docshare.wps.com/document/exam/",3,{"item":99,"name":65,"@type":92,"position":19},"https://docshare.wps.com/document/intervals-of-concavity-worksheet-04-kuta-software-infinite-calculus/522856/",{"url":99,"name":65,"@type":101,"image":102,"author":107,"headline":65,"publisher":109,"fileFormat":112,"inLanguage":73,"description":66,"dateModified":113,"datePublished":114,"encodingFormat":112,"isAccessibleForFree":115,"interactionStatistic":116},"DigitalDocument",{"url":103,"@type":104,"width":105,"height":106},"https://docshare.wps.com/thumbnails/intervals-of-concavity-worksheet-04-kuta-software-infinite-calculus/522856.png","ImageObject",300,407,{"name":63,"@type":108},"Person",{"url":90,"name":110,"@type":111},"DocShare","Organization","application/pdf","2026-10-12","2026-10-08",true,{"@type":117,"interactionType":118,"userInteractionCount":29},"InteractionCounter",{"@type":119},"ViewAction",{"@type":121,"mainEntity":122},"FAQPage",[123,129,133],{"name":124,"@type":125,"acceptedAnswer":126},"What does the worksheet ask you to find for each function?","Question",{"text":127,"@type":128},"For each problem, find the x-coordinates of all inflection points, locate any discontinuities, and determine the open intervals where the function is concave up and concave down.","Answer",{"name":130,"@type":125,"acceptedAnswer":131},"How are concavity intervals reported?",{"text":132,"@type":128},"The worksheet lists concave up intervals and concave down intervals as open intervals, written in forms such as (−∞, a), (a, b), or (a, ∞), based on where f'' changes sign.",{"name":134,"@type":125,"acceptedAnswer":135},"What is included in the critical thinking question?",{"text":136,"@type":128},"The worksheet includes a sketching task for a continuous curve y = f(x) with given values of f(1), f'(0), f'(2), and specified inequality conditions on f''(x) for x \u003C 1 and x > 1.","https://schema.org",{"og:url":99,"og:type":139,"og:title":65,"og:site_name":110,"og:description":66},"article",{"robots":141,"canonical":99},"index,follow",{"doc_id":61,"site_id":74},1791468594]