Write the function: f(x) = x3 \u2013 x2 + x \u2013<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To analyze the function ( f(x) = x^3 – x^2 + x ), we can investigate its key properties, including its roots, behavior, and critical points. Here\u2019s a step-by-step breakdown:<\/p>\n\n\n\n To find the roots of the function, we need to solve for ( f(x) = 0 ):<\/p>\n\n\n\n [ Factoring out ( x ):<\/p>\n\n\n\n [ This gives us one root at ( x = 0 ). To find the other roots, we need to solve the quadratic equation ( x^2 – x + 1 = 0 ). The discriminant ( D ) of this quadratic is calculated as follows:<\/p>\n\n\n\n [ Since the discriminant is negative, ( x^2 – x + 1 ) has no real roots. Therefore, the only real root of the function ( f(x) ) is ( x = 0 ).<\/p>\n\n\n\n Next, we can investigate the behavior of the function by calculating its first derivative:<\/p>\n\n\n\n [ To find critical points, we set ( f'(x) = 0 ):<\/p>\n\n\n\n [ Calculating the discriminant for this quadratic:<\/p>\n\n\n\n [ Again, since the discriminant is negative, there are no real critical points, indicating that ( f'(x) ) does not change sign. Since the leading coefficient of ( f'(x) ) is positive, ( f'(x) ) is always positive, meaning that the function ( f(x) ) is monotonically increasing for all ( x ).<\/p>\n\n\n\n Thus, the function ( f(x) = x^3 – x^2 + x ) has one real root at ( x = 0 ) and is increasing everywhere. As ( x ) approaches negative infinity, ( f(x) ) approaches negative infinity, and as ( x ) approaches positive infinity, ( f(x) ) approaches positive infinity. This means the function crosses the x-axis at ( x = 0 ) and has no maximum or minimum points.<\/p>\n","protected":false},"excerpt":{"rendered":" Write the function: f(x) = x3 \u2013 x2 + x \u2013 The Correct Answer and Explanation is: To analyze the function ( f(x) = x^3 – x^2 + x ), we can investigate its key properties, including its roots, behavior, and critical points. Here\u2019s a step-by-step breakdown: Step 1: Finding the Roots To find the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-157586","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/157586","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=157586"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/157586\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=157586"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=157586"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=157586"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Finding the Roots<\/h3>\n\n\n\n
x^3 – x^2 + x = 0
]<\/p>\n\n\n\n
x(x^2 – x + 1) = 0
]<\/p>\n\n\n\n
D = b^2 – 4ac = (-1)^2 – 4(1)(1) = 1 – 4 = -3
]<\/p>\n\n\n\nStep 2: Analyzing the Behavior<\/h3>\n\n\n\n
f'(x) = 3x^2 – 2x + 1
]<\/p>\n\n\n\n
3x^2 – 2x + 1 = 0
]<\/p>\n\n\n\n
D’ = (-2)^2 – 4(3)(1) = 4 – 12 = -8
]<\/p>\n\n\n\nStep 3: Conclusion<\/h3>\n\n\n\n