3 3 Example 7 Sketch the graph of the function Available in release 6 0 and prior In 1 H Needs Miscellaneous RealOnly D L Introduce the function In 2 Out 2 In 3 Clear fD f x D H6 xL H1 3L Hx H2 3LL f xD H6 xL1 3 x2 3 FullSimplify f xDD 4 x Out 3 In 4 Out 4 H6 xL2 3 x1 3 Simplify f xDD 8 H6 xL5 3 x4 3 Provide a Real Only version Mathematica now extends roots into the complex plane In 5 Out 5 In 6 Out 6 88 H1 3L H 8L H1 3L 92 2 H 1L1 3 N D 82 1 1 73205 In 7 OneThird x D Module 8 Return If x 0 H xL H1 3L x H1 3L DDD In 8 OneThird 3D Out 8 In 9 31 3 g x D OneThird 6 xD OneThird x 2D 2 Section 3p3 Example 7 nb In 10 Plot g xD 8x 2 8 PlotStyle 8Red Thick PlotLabel fHxL PlotRange 8 4 4 AspectRatio AutomaticD f HxL 4 2 Out 10 2 2 4 6 8 2 4 Intervals of increase decrease Find on which intervals the function is increasing or decreasing In 11 g xD Out 11 IfA6 x 0 H H6 xLL1 3 H6 xL1 3 E IfBx2 0 IfB6 x 0 1 3 H H6 xLL 1 3 1 2x 3 1 3 2 3 I x2 M 2 3 x Ix2 M H6 xL 1 3 F IfBx2 0 I x2 M 1 1 1 3 1 3 F Ix2 M 1 3 F Section 3p3 Example 7 nb In 12 Plot g xD 8x 1 7 PlotStyle 8Thick Red PlotRange 8 4 4 PlotLabel f HxL Exclusions 80 AspectRatio AutomaticD f HxL 4 2 Out 12 2 4 6 2 4 The value at the local maximum located at x 4 is In 13 f 4D Out 13 2 22 3 Concave up concave down Find out on which intervals the concavity does not change sign In 14 g xD Out 14 2 IfB6 x 0 1 3 H H6 xLL 1 3 1 I 1 3 M H 1L 1 IfB6 x 0 H 6 xL5 3 3 IfBx2 0 I x2 M 1 3 IfBx2 0 Ix2 M H4 xL H2 xL J3 1 5 3 I x2 M N 3 H6 xL 1 3 F IfBx2 0 1 3 1 3 1 3 H6 xL 2 3 1 1 3 2x 3 I x2 M 2 2 3 3 x Ix2 M 1 2 3 I x2 M 1 3 2 1 1 3 1 3 F H 1LF F IfA6 x 0 H H6 xLL1 3 H6 xL1 3 E 2 3 1 x Ix2 M 2 1 3 H2 xL 2 3 Ix2 M 1 1 3 F 3 4 Section 3p3 Example 7 nb In 15 Plot g xD 8x 1 7 Exclusions 86 PlotRange 8 3 3 PlotStyle 8Thick Red AspectRatio Automatic PlotLabel f HxL D f HxL 3 2 1 Out 15 2 1 2 3 4 6
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