[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-522858-105":59,"doc-detail-522858-en":130},{"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":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","intervals-of-increase-and-decrease-worksheets-04-04","Intervals of Increase and Decrease - Worksheets - 04 - 04","","Intervals of Increase and Decrease worksheets from Infinite Calculus focusing on analyzing functions using critical points and discontinuities. For each given function, the worksheet requires determining x-coordinates of critical points, listing discontinuities, and then identifying open intervals where the function is increasing and decreasing. It includes multiple polynomial examples, a rational expression with a discontinuity, and a section on critical thinking that proves how sums preserve monotonicity and asks for a counterexample involving differences.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/intervals-of-increase-and-decrease-worksheets-04-04/522858/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/intervals-of-increase-and-decrease-worksheets-04-04/522858.png","ImageObject",300,407,{"name":92,"@type":93},"Quinn Holloway","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-14","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What steps are required to solve each interval of increase/decrease problem?","Question",{"text":112,"@type":113},"Identify the x-coordinates of all critical points, find all discontinuities, then determine the open intervals where the function is increasing and where it is decreasing.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the worksheet determine whether f + g is increasing on an interval?",{"text":117,"@type":113},"It uses the fact that if x1 \u003C x2 and both f and g are increasing, then f(x1) \u003C f(x2) and g(x1) \u003C g(x2), which implies f(x1)+g(x1) \u003C f(x2)+g(x2).",{"name":119,"@type":110,"acceptedAnswer":120},"What example is given for functions that are increasing but whose difference is decreasing?",{"text":121,"@type":113},"One example stated is f = x and g = 2x, where both functions are increasing on (−∞, ∞) but f − g decreases.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},522858,1791488676,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":52},2336474466712,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Kuta Software-Infinite Calculus Name   \nIntervals of Increase and Decrease Date  Period   \nFor each problem, find the x-coordinates of all critical points, find all discontinuities, and find the open intervals where the function is increasing and decreasing.  \n1) y = −x3 + 2x2 + 2  \n2) y = x 3 − 11x2 + 39x − 47  \n3) y = −x4 + 3x2 − 3  \n4) y = 4xx~~ ~~4  \n3x2 − 3  \n5) y = ~~ ~~  \nx 3  \n2  \n6) y = (2x − 8) 3  \n5 2  \n1  3   3  \n7) y = −  5  (x − 4) − 2(x − 4) − 1  \nCritical thinking question:  \n8) If functions f and g are increasing on an interval, show that f + g is increasing on the same interval.  \n9) Give an example where functions f and g are increasing on the interval (−∞, ∞), but where f − g is decreasing.  \nKuta Software-Infinite Calculus Name   \nIntervals of Increase and Decrease Date  Period   \nFor each problem, find the x-coordinates of all critical points, find all discontinuities, and find the open intervals where the function is increasing and decreasing.  \n1) y = −x3 + 2x2 + 2  \nCritical points at: x = 0, 43 No discontinuities exist.  \nIncreasing: (0, 43) Decreasing: (−∞, 0),(~~ ~~43~~ ~~ , ∞)  \n2) y = x 3 − 11x2 + 39x − 47  \nCritical points at: x = 3, ~~ ~~133 No discontinuities exist.  \nIncreasing: (−∞, 3),(~~ ~~133~~ ~~ , ∞) Decreasing: (3, ~~ ~~133)  \n3) y = −x4 + 3x2 − 3  \nCritical points at: x = − ~~ ~~2~~ ~~, 0, ~~ ~~2~~ ~~ No discontinuities exist.  \nIncreasing: (−∞, −~~ ~~26~~ ~~ ),(0,~~ ~~26 ) Decreasing: (−~~ ~~26 , 0),(~~ ~~26 , ∞)  \n4) y = 4xx~~ ~~4  \nCritical points at: x = −2, 0 Discontinuity at: x = −1  \nIncreasing: (−∞, −2), (0, ∞) Decreasing: (−2, −1), (−1, 0)  \n3x2 − 3  \n5) y = ~~ ~~  \nx 3  \nCritical points at: x = − 3 , 3 Discontinuity at: x = 0  \nIncreasing: (− 3 , 0), (0, 3 ) Decreasing: (−∞, − 3 ), ( 3 , ∞)  \n2  \n6) y = (2x − 8) 3  \nCritical point at: x = 4 No discontinuities exist.  \nIncreasing: (4, ∞) Decreasing: (−∞, 4)  \n5 2  \n1  3   3  \n7) y = −  5  (x − 4) − 2(x − 4) − 1  \nCritical points at: x = 0, 4 No discontinuities exist.  \nIncreasing: (0, 4) Decreasing: (−∞, 0), (4, ∞)  \nCritical thinking question:  \n8) If functions f and g are increasing on an interval, show that f + g is increasing on the same interval.  \nWe know that if x1 \u003C x2 , then f (x1) \u003C f (x2) and g(x1) \u003C g(x2) . Therefore,  \nf (x1) + g(x1) \u003C f (x2) + g(x2) .  \n9) Give an example where functions f and g are increasing on the interval (−∞, ∞), but where f − g is decreasing.  \nMany answers. Ex: f = x and g = 2x  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCair29x69d1YJo","https://ap.wps.com/l/cbCair29x69d1YJo","pdf",37922,"English","# Intervals of Increase and Decrease\n## Problems 1-7\n## Critical Thinking Questions 8-9","[{\"question\":\"What steps are required to solve each interval of increase/decrease problem?\",\"answer\":\"Identify the x-coordinates of all critical points, find all discontinuities, then determine the open intervals where the function is increasing and where it is decreasing.\"},{\"question\":\"How does the worksheet determine whether f + g is increasing on an interval?\",\"answer\":\"It uses the fact that if x1 \\u003c x2 and both f and g are increasing, then f(x1) \\u003c f(x2) and g(x1) \\u003c g(x2), which implies f(x1)+g(x1) \\u003c f(x2)+g(x2).\"},{\"question\":\"What example is given for functions that are increasing but whose difference is decreasing?\",\"answer\":\"One example stated is f = x and g = 2x, where both functions are increasing on (−∞, ∞) but f − g decreases.\"}]","Intervals of Increase and Decrease - Worksheets - 04 - 04 | PDF",1791426242]