3.3 Example 7Sketch the graph of the functionAvailable in release 6.0 and prior:In[1]:=H∗ Needs@"Miscellaneous`RealOnly`"D ∗Lü Introduce the functionIn[2]:=Clear@fD;f@x_D := H6 − xL^H1 ê 3LHx^H2 ê 3LL;f@xDOut[2]=H6 − xL1ê3x2ê3In[3]:=FullSimplify@f'@xDDOut[3]=4 − xH6 − xL2ê3x1ê3In[4]:=Simplify@f''@xDDOut[4]=−8H6 − xL5ê3x4ê3ü Provide a “Real Only” versionMathematica now extends roots into the complex plane:In[5]:=88^H1 ê 3L, H− 8L^H1 ê 3L<Out[5]=92, 2 H− 1L1ê3=In[6]:=N@%DOut[6]=82., 1. + 1.73205 <In[7]:=OneThird@x_D := Module@8 <, Return @If@ x < 0,− H− xL^H1 ê 3L,x^H1 ê 3L DDDIn[8]:=OneThird@− 3DOut[8]=− 31ê3In[9]:=g@x_D := OneThird@6 − xD OneThird@x^2DIn[10]:=Plot@g@xD, 8x, − 2, 8<, PlotStyle → 8Red, Thick<,PlotLabel −> "fHxL", PlotRange −> 8−4, 4<, AspectRatio → AutomaticDOut[10]=-2 2 4 6 8-4-224fHxLü Intervals of increase/decreaseFind on which intervals the function is increasing or decreasing:In[11]:=g'@xDOut[11]=IfA6 − x < 0, − H− H6 − xLL1ê3, H6 − xL1ê3E IfBx2< 0,2x3 I−x2M2ê3,23x Ix2M−1+13F+IfB6 − x < 0, −13H− H6 − xLL−1+13, −13H6 − xL−1+13F IfBx2< 0, − I− x2M1ê3, Ix2M1ê3F2 Section 3p3 Example 7.nbIn[12]:=Plot@g'@xD, 8x, − 1, 7<, PlotStyle → 8Thick, Red<, PlotRange → 8− 4, 4<,PlotLabel → "f'HxL", Exclusions → 80<, AspectRatio → AutomaticDOut[12]=2 4 6-4-224f'HxLThe value at the local maximum located at x = 4 is:In[13]:=f@4DOut[13]=2 × 22ê3ü Concave up, concave downFind out on which intervals the concavity does not change sign:In[14]:=g''@xDOut[14]=2IfB6 − x < 0, −13H− H6 − xLL−1+13, −13H6 − xL−1+13F IfBx2< 0,2x3 I− x2M2ê3,23x Ix2M−1+13F+IfB6 − x < 0,I− 1 +13MH− 1LH− 6 + xL5ê33,13− −1 +13H6 − xL−2+13H− 1LFIfBx2< 0, − I− x2M1ê3, Ix2M1ê3F+ IfA6 − x < 0, − H− H6 − xLL1ê3, H6 − xL1ê3EIfBx2< 0,H4xLH2xLJ3 I−x2M5ê3N 3+23 I− x2M2ê3,132 − 1 +13x Ix2M−2+13H2xL +23Ix2M−1+13FSection 3p3 Example 7.nb 3In[15]:=Plot@g''@xD, 8x, − 1, 7<, Exclusions → 86<, PlotRange → 8− 3, 3<,PlotStyle → 8Thick, Red<, AspectRatio → Automatic, PlotLabel → "f''HxL"DOut[15]=2 4 6-3-2-1123f''HxL4 Section 3p3 Example
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