Para hallar los extremos hay que resolver la ecuación
dxdf(x)=0(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
dxdf(x)=primera derivada15x4cos(3x)+20x3sin(3x)=0Resolvermos esta ecuaciónRaíces de esta ecuación
x1=−91.634635567103x2=−63.3624651297861x3=44.5158768953691x4=40.3281224156889x5=−60.221238182616x6=73.8334462595067x7=12.079417026695x8=49.7508149542794x9=−36.1406075887119x10=75.9276753838356x11=64.4095487013249x12=53.9389119829474x13=−1.78467728039067x14=−2.76764306086567x15=3.77827519433957x16=−49.7508149542794x17=93.7289223191141x18=−75.9276753838356x19=−65.4566359175226x20=−21.4882064963395x21=88.4932150522796x22=31.9534263797684x23=22.5344472736599x24=−82.2104134768321x25=51.8448494812653x26=14.1684434456615x27=−67.550820606457x28=−47.6568120846336x29=80.1161596550644x30=−78.0219134150581x31=100.011809795649x32=18.3501348839456x33=20.4420631577666x34=86.3989416613461x35=−18.3501348839456x36=66.5037266063558x37=0x38=5.83447607252139x39=78.0219134150581x40=7.9096484587492x41=29.8600046009359x42=−31.9534263797684x43=−12.079417026695x44=95.8232138061131x45=−9.99259294922328x46=90.5874940693044x47=82.2104134768321x48=−51.8448494812653x49=71.7392268216346x50=−5.83447607252139x51=43.4689195888571x52=−71.7392268216346x53=−25.6736357826784x54=36.1406075887119x55=−38.234330134717x56=−23.5807726014381x57=38.234330134717x58=−27.7667291875486x59=−73.8334462595067x60=−14.1684434456615x61=−53.9389119829474x62=−13.1237193239869x63=−93.7289223191141x64=−87.4460776282981x65=−69.6450179434699x66=−43.4689195888571x67=−56.0329993263719x68=26.7201570725466x69=97.9175097243864x70=−97.9175097243864x71=−7.9096484587492x72=−95.8232138061131x73=84.3046743156037x74=34.0469676095731x75=−100.011809795649x76=58.1271088294339x77=−58.1271088294339x78=−84.3046743156037x79=−80.1161596550644x80=−3.77827519433957x81=−29.8600046009359x82=56.0329993263719x83=−35.0937763570717x84=−16.2588365740921x85=−50.7978285049785x86=−34.0469676095731x87=62.315385386479x88=60.221238182616x89=9.99259294922328x90=−45.5628452306873x91=42.4219741262856x92=−89.5403538821935x93=16.2588365740921x94=27.7667291875486x95=1.78467728039067x96=21.4882064963395Signos de extremos en los puntos:
(-91.63463556710299, 352502882.774754)
(-63.36246512978607, -80575329.3464298)
(44.5158768953691, 19626153.617958)
(40.32812241568886, 13217978.5642116)
(-60.22123818261599, 65744943.8717191)
(73.83344625950674, 148563384.43735)
(12.079417026694966, -105801.386583899)
(49.75081495427937, -30620676.205518)
(-36.14060758871191, -8524247.27036998)
(75.9276753838356, 166151202.06203)
(64.40954870132492, -86035553.1893984)
(53.93891198294738, -42310290.1652217)
(-1.7846772803906716, 48.6352438938571)
(-2.7676430608656735, -256.295044714754)
(3.7782751943395687, -952.855158790833)
(-49.75081495427937, 30620692.205518)
(93.72892231911408, -385851818.778314)
(-75.9276753838356, -166151186.06203)
(-65.45663591752259, -91768706.6093194)
(-21.4882064963395, -1063978.7911248)
(88.49321505227961, 306591860.100884)
(31.953426379768374, 5207900.12662698)
(22.53444727365992, -1287051.88610351)
(-82.21041347683207, -228360091.27821)
(51.84484948126533, -36111772.4828875)
(14.168443445661534, -200598.036368131)
(-67.550820606457, -104089735.311377)
(-47.65681208463359, 25781025.1444908)
(80.11615965506437, 205963554.862822)
(-78.02191341505808, -185256291.490013)
(100.01180979564857, -500191780.740036)
(18.350134883945593, -565428.212317855)
(20.44206315776662, -871250.559055239)
(86.39894166134607, 278581324.408907)
(-18.350134883945593, 565444.212317855)
(66.50372660635584, -97783736.8115913)
(0, 8)
(5.8344760725213884, -5640.3716615012)
(78.02191341505808, 185256307.490013)
(7.909648458749203, -19290.0895429007)
(29.860004600935945, 3970981.16650358)
(-31.953426379768374, -5207884.12662698)
(-12.079417026694966, 105817.386583899)
(95.82321380611309, -421512916.768012)
(-9.99259294922328, 49422.0765649696)
(90.58749406930437, 336663442.788779)
(82.21041347683207, 228360107.27821)
(-51.84484948126533, 36111788.4828875)
(71.73922682163465, 132410309.430682)
(-5.8344760725213884, 5656.3716615012)
(43.46891958885709, -17843538.6981349)
(-71.73922682163465, -132410293.430682)
(-25.673635782678396, -2169366.94774642)
(36.14060758871191, 8524263.27036998)
(-38.234330134716956, -10678732.5315033)
(-23.580772601438067, -1543500.34903306)
(38.234330134716956, 10678748.5315033)
(-27.766729187548563, -2968708.82293071)
(-73.83344625950674, -148563368.43735)
(-14.168443445661534, 200614.036368131)
(-53.93891198294738, 42310306.1652217)
(-13.12371932398686, -147551.724253833)
(-93.72892231911408, 385851834.778314)
(-87.44607762829814, 292335040.429235)
(-69.64501794346985, -117611724.812966)
(-43.46891958885709, 17843554.6981349)
(-56.03299932637194, 49274546.3973751)
(26.720157072546627, -2545567.64244689)
(97.9175097243864, -459590641.013344)
(-97.9175097243864, 459590657.013344)
(-7.909648458749203, 19306.0895429007)
(-95.82321380611309, 421512932.768012)
(84.30467431560369, 252535436.790166)
(34.046967609573116, 6713538.80770276)
(-100.01180979564857, 500191796.740036)
(58.1271088294339, -57065103.2804346)
(-58.1271088294339, 57065119.2804346)
(-84.30467431560369, -252535420.790166)
(-80.11615965506437, -205963538.862822)
(-3.7782751943395687, 968.855158790833)
(-29.860004600935945, -3970965.16650358)
(56.03299932637194, -49274530.3973751)
(-35.093776357071675, 7578402.24394453)
(-16.258836574092104, 348243.000943114)
(-50.79782850497848, -33281350.2965328)
(-34.046967609573116, -6713522.80770276)
(62.31538538647896, -75379232.1847793)
(60.22123818261599, -65744927.8717191)
(9.99259294922328, -49406.0765649696)
(-45.56284523068725, 21539104.0674156)
(42.421974126285555, 16185246.2978379)
(-89.54035388219354, 321363905.444071)
(16.258836574092104, -348227.000943114)
(27.766729187548563, 2968724.82293071)
(1.7846772803906716, -32.6352438938571)
(21.4882064963395, 1063994.7911248)
Intervalos de crecimiento y decrecimiento de la función:Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=−63.3624651297861x2=12.079417026695x3=49.7508149542794x4=−36.1406075887119x5=64.4095487013249x6=53.9389119829474x7=−2.76764306086567x8=3.77827519433957x9=93.7289223191141x10=−75.9276753838356x11=−65.4566359175226x12=−21.4882064963395x13=22.5344472736599x14=−82.2104134768321x15=51.8448494812653x16=14.1684434456615x17=−67.550820606457x18=−78.0219134150581x19=100.011809795649x20=18.3501348839456x21=20.4420631577666x22=66.5037266063558x23=5.83447607252139x24=7.9096484587492x25=−31.9534263797684x26=95.8232138061131x27=43.4689195888571x28=−71.7392268216346x29=−25.6736357826784x30=−38.234330134717x31=−23.5807726014381x32=−27.7667291875486x33=−73.8334462595067x34=−13.1237193239869x35=−69.6450179434699x36=26.7201570725466x37=97.9175097243864x38=58.1271088294339x39=−84.3046743156037x40=−80.1161596550644x41=−29.8600046009359x42=56.0329993263719x43=−50.7978285049785x44=−34.0469676095731x45=62.315385386479x46=60.221238182616x47=9.99259294922328x48=16.2588365740921x49=1.78467728039067Puntos máximos de la función:
x49=−91.634635567103x49=44.5158768953691x49=40.3281224156889x49=−60.221238182616x49=73.8334462595067x49=75.9276753838356x49=−1.78467728039067x49=−49.7508149542794x49=88.4932150522796x49=31.9534263797684x49=−47.6568120846336x49=80.1161596550644x49=86.3989416613461x49=−18.3501348839456x49=78.0219134150581x49=29.8600046009359x49=−12.079417026695x49=−9.99259294922328x49=90.5874940693044x49=82.2104134768321x49=−51.8448494812653x49=71.7392268216346x49=−5.83447607252139x49=36.1406075887119x49=38.234330134717x49=−14.1684434456615x49=−53.9389119829474x49=−93.7289223191141x49=−87.4460776282981x49=−43.4689195888571x49=−56.0329993263719x49=−97.9175097243864x49=−7.9096484587492x49=−95.8232138061131x49=84.3046743156037x49=34.0469676095731x49=−100.011809795649x49=−58.1271088294339x49=−3.77827519433957x49=−35.0937763570717x49=−16.2588365740921x49=−45.5628452306873x49=42.4219741262856x49=−89.5403538821935x49=27.7667291875486x49=21.4882064963395Decrece en los intervalos
[100.011809795649,∞)Crece en los intervalos
(−∞,−84.3046743156037]