L4 Concavity and Points of Inflection - Part 5
Completion requirements
Unit 3
Curve Sketching
Lesson 4: Concavity and Points of Inflection
The Second Derivative Test
The second derivative test can be used to determine whether a point is a local minimum or a local maximum by investigating the concavity of the curve at that point.If «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»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» and
- «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»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#60;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math», a relative maximum occurs (the curve is concave down).
- «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«mo mathvariant=¨italic¨»``«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#62;«/mo»«mn»0«/mn»«/mstyle»«/math», a relative minimum occurs (the curve is concave up).
- «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»f«/mi»«mo mathvariant=¨italic¨»``«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»0«/mn»«/mstyle»«/math», the Second Derivative Test fails.
For each of the following functions, use the Second Derivative Test to determine the local maximum and minimum values.
a.
«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»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math»
b.
«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»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi
mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math»
a.
Find the first derivative and the critical point(s).
The critical point is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»7«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
Now, find the second derivative at this point.
Since «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»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» for all «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/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»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»7«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» represents a minimum.
As shown below on the 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»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»6«/mn»«mi»x«/mi»«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»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»7«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» is a local and an absolute minimum.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo
mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»6«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»6«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»6«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»3«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mn»3«/mn»«/mfenced»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»6«/mn»«mfenced»«mn»3«/mn»«/mfenced»«mo»+«/mo»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»9«/mn»«mo»§#8722;«/mo»«mn»18«/mn»«mo»+«/mo»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»7«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mn»3«/mn»«/mfenced»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»6«/mn»«mfenced»«mn»3«/mn»«/mfenced»«mo»+«/mo»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»9«/mn»«mo»§#8722;«/mo»«mn»18«/mn»«mo»+«/mo»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»7«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The critical point is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»7«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
Now, find the second derivative at this point.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»6«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»``«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo
mathvariant=¨italic¨»``«/mo»«mi mathvariant=¨normal¨»(«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Since «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»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#62;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» for all «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/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»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»7«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» represents a minimum.
As shown below on the 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»«mn»2«/mn»«/msup»«mo»§#8722;«/mo»«mn»6«/mn»«mi»x«/mi»«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»3«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8722;«/mo»«mn»7«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» is a local and an absolute minimum.
b.
Find the first derivative and the critical point(s).
The critical point is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
Now, find the second derivative at this point.
Since «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»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math», the Second Derivative Test fails.
Since the Second Derivative Test failed, test the slope of the function (the derivative) on either side of the critical point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math».
Because the function is increasing on the left of the critical point and decreasing on the right of the critical point, the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» is a maximum point (local and absolute), as shown below in the graph of the function «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»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»3«/mn»«/msup»«mo»§#8729;«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»3«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»3«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mo»-«/mo»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»1«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The critical point is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
Now, find the second derivative at this point.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»3«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo
mathvariant=¨italic¨»``«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»12«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi
mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»``«/mo»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»12«/mn»«mi
mathvariant=¨normal¨»(«/mi»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Since «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»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math», the Second Derivative Test fails.
Since the Second Derivative Test failed, test the slope of the function (the derivative) on either side of the critical point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»3«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo
mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»4«/mn»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»3«/mn»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»3«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»4«/mn»«mi
mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
| Since «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»§#62;«/mo»«mn»0«/mn»«/math», the function is increasing on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»-«/mo»«mo»§#8734;«/mo»«mo»,«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math». |
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»4«/mn»«msup»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mn»3«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»4«/mn»«mi
mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»4«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
| Since «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»§#60;«/mo»«mn»0«/mn»«/math», the function is decreasing on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»-«/mo»«mn»2«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math». |
Because the function is increasing on the left of the critical point and decreasing on the right of the critical point, the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» is a maximum point (local and absolute), as shown below in the graph of the function «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»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«msup»«mi mathvariant=¨normal¨»)«/mi»«mn»4«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».