L3 Maximum and Minimum Values - Part 3
Completion requirements
Unit 3
Curve Sketching
Lesson 3: Maximum and Minimum Values
Maximum and minimum points often occur where the slope of a curve changes sign.
A maximum occurs when a continuous function changes from increasing to decreasing.
A minimum occurs when a continuous function changes from decreasing to increasing.
Notice that the tangent line would be horizontal at both the maximum and the minimum points. As discussed in Lesson 2: Increasing and Decreasing Functions, where the slope of the tangent line is zero or undefined, a critical point occurs. As such, it continues to be important to find critical points: to determine intervals of increase and decrease and to locate maximum and minimum points.
If the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» is defined on the interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»a«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»b«/mi»«/mrow»«/mfenced»«/mstyle»«/math» and has a minimum or maximum value at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mi»c«/mi»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»a«/mi»«mo»§#60;«/mo»«mi»c«/mi»«mo»§#60;«/mo»«mi»b«/mi»«/mrow»«/mstyle»«/math», and if the derivative «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»c«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» exists and is finite, then «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»c«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math». Many continuous functions, such as polynomial functions, behave like this.
Watch the video Extrema Local and Absolute Maximums and Minimums. Please note the video discusses the second derivative and concavity. These concepts will be explored in the next Lesson - Lesson 4: Concavity and Points of Inflection.
Note: Different resources define local extrema differently. The video Extrema Local and Absolute Maximums and Minimums includes the possibility of LOCAL extrema occurring at the closed endpoints of a function. This is not a common definition. As such, for the purposes of this course, the ONLY extrema that may occur at the endpoints will be ABSOLUTE extrema.
A maximum occurs when a continuous function changes from increasing to decreasing.
A minimum occurs when a continuous function changes from decreasing to increasing.
Notice that the tangent line would be horizontal at both the maximum and the minimum points. As discussed in Lesson 2: Increasing and Decreasing Functions, where the slope of the tangent line is zero or undefined, a critical point occurs. As such, it continues to be important to find critical points: to determine intervals of increase and decrease and to locate maximum and minimum points.
If the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» is defined on the interval «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»a«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»b«/mi»«/mrow»«/mfenced»«/mstyle»«/math» and has a minimum or maximum value at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mi»c«/mi»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»a«/mi»«mo»§#60;«/mo»«mi»c«/mi»«mo»§#60;«/mo»«mi»b«/mi»«/mrow»«/mstyle»«/math», and if the derivative «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»c«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» exists and is finite, then «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»c«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math». Many continuous functions, such as polynomial functions, behave like this.
Watch the video Extrema Local and Absolute Maximums and Minimums. Please note the video discusses the second derivative and concavity. These concepts will be explored in the next Lesson - Lesson 4: Concavity and Points of Inflection.
Note: Different resources define local extrema differently. The video Extrema Local and Absolute Maximums and Minimums includes the possibility of LOCAL extrema occurring at the closed endpoints of a function. This is not a common definition. As such, for the purposes of this course, the ONLY extrema that may occur at the endpoints will be ABSOLUTE extrema.