[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-522845-105":59,"doc-detail-522845-en":122},{"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":115,"head_meta":117,"extra_data":119,"updated_unix":121},105,"en","absolute-extrema","Absolute Extrema","","Worksheet on finding absolute minima and absolute maxima for a set of functions over specified intervals. It provides multiple polynomial and transformed cubic/quartic expressions, each paired with a closed or open interval of x values. For every problem, the task is to determine the exact x-location(s) and corresponding y-values of the absolute extreme points, or identify when an absolute maximum or minimum does not exist on the given domain.",{"@graph":69,"@context":114},[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/absolute-extrema/522845/",{"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/absolute-extrema/522845.png","ImageObject",300,407,{"name":92,"@type":93},"Taylor Morgan","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-12","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108],{"name":109,"@type":110,"acceptedAnswer":111},"What does it mean if the worksheet says there is “no absolute maximum” or “no absolute minimum”?","Question",{"text":112,"@type":113},"It means the function does not reach a highest (or lowest) value on the given interval, often due to the interval being open or the function being unbounded there.","Answer","https://schema.org",{"og:url":83,"og:type":116,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":118,"canonical":83},"index,follow",{"doc_id":120,"site_id":62},522845,1791497823,{"code":4,"msg":5,"data":123},{"doc_id":120,"user_id":124,"nickname":92,"user_avatar":125,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":126,"file_id":127,"file_url":128,"file_type":129,"file_size":130,"view_count":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":131,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":132,"faqs":133,"seo_title":134,"seo_description":67,"update_tm":135,"read_time":52},1099523885336,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Kuta Software-Infinite Calculus Name   \nAbsolute Extrema Date  Period   \nFor each problem, find all points of absolute minima and maxima on the given closed interval.  \n1) y = − x 3 − 6x 2 − 9x + 3; [−3, −1]  \n2) y = x~~ ~~2~~ ~~~~ ~~4~~ ~~ ; [0, 5]  \n3) y = x 3 + 6x 2 + 9x + 3; [−4, 0] 4) y = x 4 − 3x 2 + 4; [−1, 1]  \n2 2  \n5) y = ~~ ~~3xx−~~ ~~6~~ ~~ ; [3, 6] 6) y = (x + 2) 3  ; [−4, −2]  \nFor each problem, find all points of absolute minima and maxima on the given interval.  \n7) y = x 3 − 3x 2 − 3; (0, 3)  \n1   \n8) y = (5x + 25) 3 ; [−2, 2]  \n9) y = x 3 − 3x 2 + 6; [0, ∞) 10) y = x 4 − 2x 2 − 3; (0, ∞)  \n4  7   1  \n11) y = x 2 + 2  ; (−5, −2] 12) y = − ~~ ~~16~~ ~~ (x + 1) 3 + ~~ ~~143~~ ~~ (x + 1) 3 ; (−5, 0)  \nKuta Software-Infinite Calculus Name   \nAbsolute Extrema Date  Period   \nFor each problem, find all points of absolute minima and maxima on the given closed interval.  \n1) y = − x 3 − 6x 2 − 9x + 3; [−3, −1]  \nAbsolute minimum: (−3, 3)  \nAbsolute maximum: (−1, 7)  \n3) y = x 3 + 6x 2 + 9x + 3; [−4, 0]  \nAbsolute minima: (−4, −1), (−1, −1)  \nAbsolute maxima: (0, 3), (−3, 3)  \n2  \n5) y = ~~ ~~3xx−~~ ~~6~~ ~~ ; [3, 6]  \nAbsolute minimum: (4, ~~ ~~83)  \nAbsolute maxima: (3, 3), (6, 3)  \n2) y = x~~ ~~2~~ ~~~~ ~~4~~ ~~ ; [0, 5]  \nAbsolute minimum: (5, ~~ ~~829)  \nAbsolute maximum: (0, 2)  \n4) y = x 4 − 3x 2 + 4; [−1, 1]  \nAbsolute minima: (−1, 2), ( 1, 2)  \nAbsolute maximum: (0, 4)  \n2  \n6) y = (x + 2) 3 ; [−4, −2]  \nAbsolute minimum: (−2, 0)  \nAbsolute maximum: (−4, 3 4 )  \nFor each problem, find all points of absolute minima and maxima on the given interval.  \n7) y = x 3 − 3x 2 − 3; (0, 3)  \nAbsolute minimum: (2, −7) No absolute maxima.  \n 1   \n8) y = (5x + 25) 3 ; [−2, 2]  \nAbsolute minimum: (−2, 3 15 )  \nAbsolute maximum: (2, 3 35 )  \n9) y = x 3 − 3x 2 + 6; [0, ∞)  \nAbsolute minimum: (2, 2) No absolute maxima.  \n11) y = x~~ ~~2~~ ~~~~ ~~2~~ ~~ ; (−5, −2]  \nNo absolute minima.  \nAbsolute maximum: (−2, ~~ ~~23)  \n10) y = x 4 − 2x 2 − 3; (0, ∞)  \nAbsolute minimum: ( 1, −4)  \nNo absolute maxima.  \n7 1  \n12) y = − ~~ ~~16~~ ~~ (x + 1) 3 + ~~ ~~143~~ ~~ (x + 1) 3  ; (−5, 0)  \nAbsolute minimum: (−3, −4 3 2 ) No absolute maxima.  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCailjTd5w9tnOF","https://ap.wps.com/l/cbCailjTd5w9tnOF","pdf",42395,"English","# Absolute Extrema\n## Problems on Absolute Minima and Maxima","[{\"question\":\"What does it mean if the worksheet says there is “no absolute maximum” or “no absolute minimum”?\",\"answer\":\"It means the function does not reach a highest (or lowest) value on the given interval, often due to the interval being open or the function being unbounded there.\"}]","Absolute Extrema | PDF",1791426228]