L4 Concavity and Points of Inflection - Part 3
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Unit 3
Curve Sketching
Lesson 4: Concavity and Points of Inflection
A point on a curve is called an inflection point if the curve changes from concave up to concave down or vice versa. It is important to note that the function must be continuous at that point.
In the following graph of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/msup»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math», the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» is an inflection point as the curve changes from concave up to concave down at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
In the following graph of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/msup»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math», the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» is an inflection point as the curve changes from concave up to concave down at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»0«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
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For more information on how to draw the first derivative from the original function, and then how to draw the second derivative from the first derivative, click the Skill Builder button to access the Skill Builder page.