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Trazado el gráfico de la función/ cos(4*x+x)/(x^2-1)

Gráfico de la función y = cos(4*x+x)/(x^2-1)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(4*x + x)
f(x) = ------------
           2       
          x  - 1
f(x)=cos⁡(x+4x)x2−1f{\left(x \right)} = \frac{\cos{\left(x + 4 x \right)}}{x^{2} - 1} 
f = cos(x + 4*x)/(x^2 - 1)
Gráfico de la función
02468-8-6-4-2-10105-5
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=−1x_{1} = -1
x2=1x_{2} = 1
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
cos⁡(x+4x)x2−1=0\frac{\cos{\left(x + 4 x \right)}}{x^{2} - 1} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π10x_{1} = \frac{\pi}{10}
x2=3π10x_{2} = \frac{3 \pi}{10}
Solución numérica
x1=−29.845130209103x_{1} = -29.845130209103
x2=−70.0575161750524x_{2} = -70.0575161750524
x3=54.3495529071034x_{3} = 54.3495529071034
x4=24.1902634326414x_{4} = 24.1902634326414
x5=65.0309679293087x_{5} = 65.0309679293087
x6=27.9601746169492x_{6} = 27.9601746169492
x7=−92.0486647501809x_{7} = -92.0486647501809
x8=−97.7035315266426x_{8} = -97.7035315266426
x9=−41.7831822927443x_{9} = -41.7831822927443
x10=−85.7654794430014x_{10} = -85.7654794430014
x11=38.0132711084365x_{11} = 38.0132711084365
x12=300.022098417825x_{12} = 300.022098417825
x13=−4.08407044966673x_{13} = -4.08407044966673
x14=−75.712382951514x_{14} = -75.712382951514
x15=−61.8893752757189x_{15} = -61.8893752757189
x16=−27.9601746169492x_{16} = -27.9601746169492
x17=32.3584043319749x_{17} = 32.3584043319749
x18=2.19911485751286x_{18} = 2.19911485751286
x19=−71.9424717672063x_{19} = -71.9424717672063
x20=−14.1371669411541x_{20} = -14.1371669411541
x21=−31.7300858012569x_{21} = -31.7300858012569
x22=92.0486647501809x_{22} = 92.0486647501809
x23=48.0663675999238x_{23} = 48.0663675999238
x24=81.9955682586936x_{24} = 81.9955682586936
x25=−53.7212343763855x_{25} = -53.7212343763855
x26=−63.7743308678728x_{26} = -63.7743308678728
x27=66.2876049907446x_{27} = 66.2876049907446
x28=61.8893752757189x_{28} = 61.8893752757189
x29=71.9424717672063x_{29} = 71.9424717672063
x30=−9.73893722612836x_{30} = -9.73893722612836
x31=−5.96902604182061x_{31} = -5.96902604182061
x32=93.9336203423348x_{32} = 93.9336203423348
x33=34.2433599241287x_{33} = 34.2433599241287
x34=−80.1106126665397x_{34} = -80.1106126665397
x35=−39.8982267005904x_{35} = -39.8982267005904
x36=76.340701482232x_{36} = 76.340701482232
x37=−65.6592864600267x_{37} = -65.6592864600267
x38=−81.9955682586936x_{38} = -81.9955682586936
x39=98.3318500573605x_{39} = 98.3318500573605
x40=17.9070781254618x_{40} = 17.9070781254618
x41=−26.0752190247953x_{41} = -26.0752190247953
x42=22.3053078404875x_{42} = 22.3053078404875
x43=−11.6238928182822x_{43} = -11.6238928182822
x44=−663.818527703523x_{44} = -663.818527703523
x45=0.314159265358979x_{45} = 0.314159265358979
x46=44.2964564156161x_{46} = 44.2964564156161
x47=192.579629665054x_{47} = 192.579629665054
x48=49.9513231920777x_{48} = 49.9513231920777
x49=78.2256570743859x_{49} = 78.2256570743859
x50=47.4380490692059x_{50} = 47.4380490692059
x51=−73.8274273593601x_{51} = -73.8274273593601
x52=100.216805649514x_{52} = 100.216805649514
x53=−48.0663675999238x_{53} = -48.0663675999238
x54=−83.8805238508475x_{54} = -83.8805238508475
x55=−36.1283155162826x_{55} = -36.1283155162826
x56=60.0044196835651x_{56} = 60.0044196835651
x57=10.3672557568463x_{57} = 10.3672557568463
x58=12.2522113490002x_{58} = 12.2522113490002
x59=88.2787535658732x_{59} = 88.2787535658732
x60=36.1283155162826x_{60} = 36.1283155162826
x61=−87.6504350351552x_{61} = -87.6504350351552
x62=−68.8008791136165x_{62} = -68.8008791136165
x63=21.6769893097696x_{63} = 21.6769893097696
x64=70.0575161750524x_{64} = 70.0575161750524
x65=16.0221225333079x_{65} = 16.0221225333079
x66=−2.82743338823081x_{66} = -2.82743338823081
x67=−93.9336203423348x_{67} = -93.9336203423348
x68=4.08407044966673x_{68} = 4.08407044966673
x69=14.1371669411541x_{69} = 14.1371669411541
x70=−38.0132711084365x_{70} = -38.0132711084365
x71=−17.9070781254618x_{71} = -17.9070781254618
x72=39.8982267005904x_{72} = 39.8982267005904
x73=90.1637091580271x_{73} = 90.1637091580271
x74=−76.340701482232x_{74} = -76.340701482232
x75=5.96902604182061x_{75} = 5.96902604182061
x76=46.18141200777x_{76} = 46.18141200777
x77=−49.9513231920777x_{77} = -49.9513231920777
x78=−51.8362787842316x_{78} = -51.8362787842316
x79=−43.6681378848981x_{79} = -43.6681378848981
x80=−95.8185759344887x_{80} = -95.8185759344887
x81=80.1106126665397x_{81} = 80.1106126665397
x82=68.1725605828985x_{82} = 68.1725605828985
x83=−7.85398163397448x_{83} = -7.85398163397448
x84=−60.0044196835651x_{84} = -60.0044196835651
x85=−21.6769893097696x_{85} = -21.6769893097696
x86=−33.6150413934108x_{86} = -33.6150413934108
x87=83.8805238508475x_{87} = 83.8805238508475
x88=−16.0221225333079x_{88} = -16.0221225333079
x89=−58.1194640914112x_{89} = -58.1194640914112
x90=58.1194640914112x_{90} = 58.1194640914112
x91=7.22566310325652x_{91} = 7.22566310325652
x92=−19.7920337176157x_{92} = -19.7920337176157
x93=133.517687777566x_{93} = 133.517687777566
x94=56.2345084992573x_{94} = 56.2345084992573
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en cos(4*x + x)/(x^2 - 1).
cos⁡(0⋅4)−1+02\frac{\cos{\left(0 \cdot 4 \right)}}{-1 + 0^{2}}
Resultado:
f(0)=−1f{\left(0 \right)} = -1
Punto:
(0, -1)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
−2xcos⁡(x+4x)(x2−1)2−5sin⁡(x+4x)x2−1=0- \frac{2 x \cos{\left(x + 4 x \right)}}{\left(x^{2} - 1\right)^{2}} - \frac{5 \sin{\left(x + 4 x \right)}}{x^{2} - 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=43.9804772707051x_{1} = 43.9804772707051
x2=−43.9804772707051x_{2} = -43.9804772707051
x3=−39.5820450949106x_{3} = -39.5820450949106
x4=−69.7422096048752x_{4} = -69.7422096048752
x5=−47.7505322647089x_{5} = -47.7505322647089
x6=−89.8486594021155x_{6} = -89.8486594021155
x7=60.9455844986464x_{7} = 60.9455844986464
x8=−27.6431177325273x_{8} = -27.6431177325273
x9=96.1319029246777x_{9} = 96.1319029246777
x10=−10.0450569900272x_{10} = -10.0450569900272
x11=10.0450569900272x_{11} = 10.0450569900272
x12=77.9104708292441x_{12} = 77.9104708292441
x13=−77.9104708292441x_{13} = -77.9104708292441
x14=−59.688919779734x_{14} = -59.688919779734
x15=−5.64024876457399x_{15} = -5.64024876457399
x16=−56.5472525996449x_{16} = -56.5472525996449
x17=−76.0254897658437x_{17} = -76.0254897658437
x18=−54.0339126098678x_{18} = -54.0339126098678
x19=45.8655077270332x_{19} = 45.8655077270332
x20=65.9722328301909x_{20} = 65.9722328301909
x21=55.91891815781x_{21} = 55.91891815781
x22=−93.6186064536815x_{22} = -93.6186064536815
x23=−25.7579494846518x_{23} = -25.7579494846518
x24=−52.7772402600661x_{24} = -52.7772402600661
x25=−86.0787092076392x_{25} = -86.0787092076392
x26=70.3705383836538x_{26} = 70.3705383836538
x27=−23.8727474905771x_{27} = -23.8727474905771
x28=−79.795450688682x_{28} = -79.795450688682
x29=72.2555236499826x_{29} = 72.2555236499826
x30=−3.74702407259747x_{30} = -3.74702407259747
x31=−49.6355515592029x_{31} = -49.6355515592029
x32=−74.140507406587x_{32} = -74.140507406587
x33=−21.3590767768119x_{33} = -21.3590767768119
x34=28.271500822307x_{34} = 28.271500822307
x35=52.1489034467125x_{35} = 52.1489034467125
x36=−98.016874525578x_{36} = -98.016874525578
x37=−20.1022039761978x_{37} = -20.1022039761978
x38=−32.0417460195949x_{38} = -32.0417460195949
x39=84.1937327998049x_{39} = 84.1937327998049
x40=54.0339126098678x_{40} = 54.0339126098678
x41=64.0872415470646x_{41} = 64.0872415470646
x42=3.74702407259747x_{42} = 3.74702407259747
x43=26.3863422909371x_{43} = 26.3863422909371
x44=0x_{44} = 0
x45=32.0417460195949x_{45} = 32.0417460195949
x46=8.15819343440073x_{46} = 8.15819343440073
x47=42.0954400989517x_{47} = 42.0954400989517
x48=−81.680429427562x_{48} = -81.680429427562
x49=38.3253416359477x_{49} = 38.3253416359477
x50=16.3313658042246x_{50} = 16.3313658042246
x51=23.8727474905771x_{51} = 23.8727474905771
x52=62.2022480991767x_{52} = 62.2022480991767
x53=−65.9722328301909x_{53} = -65.9722328301909
x54=74.140507406587x_{54} = 74.140507406587
x55=11.931302161923x_{55} = 11.931302161923
x56=98.6451982565458x_{56} = 98.6451982565458
x57=−45.8655077270332x_{57} = -45.8655077270332
x58=−57.8039204446587x_{58} = -57.8039204446587
x59=76.0254897658437x_{59} = 76.0254897658437
x60=86.0787092076392x_{60} = 86.0787092076392
x61=40.2103952651705x_{61} = 40.2103952651705
x62=89.2203345987252x_{62} = 89.2203345987252
x63=18.2168332867871x_{63} = 18.2168332867871
x64=−42.0954400989517x_{64} = -42.0954400989517
x65=−37.6969882430057x_{65} = -37.6969882430057
x66=21.9875030069394x_{66} = 21.9875030069394
x67=82.3087554381567x_{67} = 82.3087554381567
x68=6.27011200881177x_{68} = 6.27011200881177
x69=−13.8171889415972x_{69} = -13.8171889415972
x70=−8.78724240122631x_{70} = -8.78724240122631
x71=50.2638902610058x_{71} = 50.2638902610058
x72=30.1566338943204x_{72} = 30.1566338943204
x73=94.2469306832098x_{73} = 94.2469306832098
x74=−99.9018455221845x_{74} = -99.9018455221845
x75=−55.91891815781x_{75} = -55.91891815781
x76=246.928858587175x_{76} = 246.928858587175
x77=−87.9636847230099x_{77} = -87.9636847230099
x78=−11.931302161923x_{78} = -11.931302161923
x79=−67.8572221291202x_{79} = -67.8572221291202
x80=87.9636847230099x_{80} = 87.9636847230099
x81=−21.9875030069394x_{81} = -21.9875030069394
x82=−1.82420087382986x_{82} = -1.82420087382986
x83=−33.9268407028905x_{83} = -33.9268407028905
x84=−35.8119207078091x_{84} = -35.8119207078091
x85=20.1022039761978x_{85} = 20.1022039761978
x86=98.016874525578x_{86} = 98.016874525578
x87=−91.7336332965327x_{87} = -91.7336332965327
x88=60.3172522833615x_{88} = 60.3172522833615
x89=−31.413377404844x_{89} = -31.413377404844
x90=−71.6271954015107x_{90} = -71.6271954015107
x91=92.3619577619375x_{91} = 92.3619577619375
x92=4.37901497732045x_{92} = 4.37901497732045
x93=−96.1319029246777x_{93} = -96.1319029246777
x94=−64.0872415470646x_{94} = -64.0872415470646
x95=48.3788725817987x_{95} = 48.3788725817987
x96=−15.7028490207672x_{96} = -15.7028490207672
x97=33.9268407028905x_{97} = 33.9268407028905
x98=99.9018455221845x_{98} = 99.9018455221845
Signos de extremos en los puntos:
(43.9804772707051, 0.000517233597437444)

(-43.9804772707051, 0.000517233597437444)

(-39.58204509491064, -0.00063864368718547)

(-69.74220960487521, -0.00020563202187301)

(-47.750532264708944, 0.000438751715954693)

(-89.84865940211554, -0.000123887158728359)

(60.94558449864636, -0.000269291773823358)

(-27.643117732527344, 0.00131023466726553)

(96.13190292467768, -0.000108220156705173)

(-10.0450569900272, 0.0100016064521053)

(10.0450569900272, 0.0100016064521053)

(77.91047082924406, 0.000164768493052767)

(-77.91047082924406, 0.000164768493052767)

(-59.688919779734, -0.000280753202870758)

(-5.640248764573985, -0.0323678101092314)

(-56.54725259964493, 0.000312825357865791)

(-76.02548976584366, -0.000173041662620807)

(-54.03391260986782, 0.000342613154727243)

(45.86550772703323, -0.000475573400322225)

(65.97223283019088, -0.000229810275503831)

(55.91891815780999, -0.000319897076825524)

(-93.61860645368154, -0.000114109353609796)

(-25.75794948465181, -0.00150931531853117)

(-52.77724026006608, 0.000359128695556195)

(-86.07870920763919, -0.000134977827793602)

(70.37053838365381, 0.000201975616905833)

(-23.872747490577076, 0.00175750566782006)

(-79.79545068868202, -0.000157074790825559)

(72.25552364998259, -0.000191573057061236)

(-3.7470240725974677, 0.0761844297909742)

(-49.635551559202874, -0.000406047177822934)

(-74.14050740658705, 0.000181953990540541)

(-21.359076776811886, 0.00219640070375977)

(28.27150082230704, -0.0012525711293605)

(52.148903446712495, -0.00036783798851984)

(-98.01687452557796, 0.000104097401776241)

(-20.102203976197767, 0.00248028917179634)

(-32.04174601959489, -0.000974893031047041)

(84.19373279980488, 0.000141090196683047)

(54.03391260986782, 0.000342613154727243)

(64.08724154706462, 0.000243530932716317)

(3.7470240725974677, 0.0761844297909742)

(26.386342290937076, 0.00143818845103276)

(0, -1)

(32.04174601959489, -0.000974893031047041)

(8.158193434400726, -0.0152352419958773)

(42.0954400989517, -0.000564618905910019)

(-81.68042942756199, 0.000149907674194891)

(38.3253416359477, -0.000681239810536001)

(16.33136580422461, 0.00376231502619645)

(23.872747490577076, 0.00175750566782006)

(62.20224809917666, -0.000258518194649307)

(-65.97223283019088, -0.000229810275503831)

(74.14050740658705, 0.000181953990540541)

(11.931302161923014, -0.00707030999095896)

(98.64519825654583, -0.000102775396689447)

(-45.86550772703323, -0.000475573400322225)

(-57.80392044465874, 0.000299367743549506)

(76.02548976584366, -0.000173041662620807)

(86.07870920763919, -0.000134977827793602)

(40.21039526517046, 0.000618828755412424)

(89.22033459872519, 0.000125638429866559)

(18.216833286787086, -0.0030217574355397)

(-42.0954400989517, -0.000564618905910019)

(-37.69698824300569, 0.000704154452351808)

(21.987503006939445, -0.00207240808303087)

(82.30875543815671, -0.000147627379267843)

(6.270112008811766, 0.0260441728434178)

(-13.817188941597209, 0.00526329323148144)

(-8.787242401226308, 0.0131067378246422)

(50.26389026100576, 0.000395955129155678)

(30.156633894320375, 0.00110071225151677)

(94.24693068320978, 0.000112592782692442)

(-99.90184552218453, -0.000100205835347264)

(-55.91891815780999, -0.000319897076825524)

(246.92885858717477, -1.64007177650704e-5)

(-87.9636847230099, 0.00012925424447455)

(-11.931302161923014, -0.00707030999095896)

(-67.85722212912023, 0.000217217407091631)

(87.9636847230099, 0.00012925424447455)

(-21.987503006939445, -0.00207240808303087)

(-1.824200873829856, -0.409937233429038)

(-33.92684070289054, 0.000869481605881122)

(-35.81192070780913, -0.000780290654154347)

(20.102203976197767, 0.00248028917179634)

(98.01687452557796, 0.000104097401776241)

(-91.7336332965327, 0.000118847566237247)

(60.31725228336151, 0.000274932913719903)

(-31.413377404843953, 0.00101432186168238)

(-71.62719540151069, 0.000194949439564309)

(92.36195776193752, -0.000117235888417475)

(4.379014977320452, -0.0547645768509519)

(-96.13190292467768, -0.000108220156705173)

(-64.08724154706462, 0.000243530932716317)

(48.37887258179868, -0.000427424341602152)

(-15.702849020767196, -0.00407067042074794)

(33.92684070289054, 0.000869481605881122)

(99.90184552218453, -0.000100205835347264)


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=−39.5820450949106x_{1} = -39.5820450949106
x2=−69.7422096048752x_{2} = -69.7422096048752
x3=−89.8486594021155x_{3} = -89.8486594021155
x4=60.9455844986464x_{4} = 60.9455844986464
x5=96.1319029246777x_{5} = 96.1319029246777
x6=−59.688919779734x_{6} = -59.688919779734
x7=−5.64024876457399x_{7} = -5.64024876457399
x8=−76.0254897658437x_{8} = -76.0254897658437
x9=45.8655077270332x_{9} = 45.8655077270332
x10=65.9722328301909x_{10} = 65.9722328301909
x11=55.91891815781x_{11} = 55.91891815781
x12=−93.6186064536815x_{12} = -93.6186064536815
x13=−25.7579494846518x_{13} = -25.7579494846518
x14=−86.0787092076392x_{14} = -86.0787092076392
x15=−79.795450688682x_{15} = -79.795450688682
x16=72.2555236499826x_{16} = 72.2555236499826
x17=−49.6355515592029x_{17} = -49.6355515592029
x18=28.271500822307x_{18} = 28.271500822307
x19=52.1489034467125x_{19} = 52.1489034467125
x20=−32.0417460195949x_{20} = -32.0417460195949
x21=0x_{21} = 0
x22=32.0417460195949x_{22} = 32.0417460195949
x23=8.15819343440073x_{23} = 8.15819343440073
x24=42.0954400989517x_{24} = 42.0954400989517
x25=38.3253416359477x_{25} = 38.3253416359477
x26=62.2022480991767x_{26} = 62.2022480991767
x27=−65.9722328301909x_{27} = -65.9722328301909
x28=11.931302161923x_{28} = 11.931302161923
x29=98.6451982565458x_{29} = 98.6451982565458
x30=−45.8655077270332x_{30} = -45.8655077270332
x31=76.0254897658437x_{31} = 76.0254897658437
x32=86.0787092076392x_{32} = 86.0787092076392
x33=18.2168332867871x_{33} = 18.2168332867871
x34=−42.0954400989517x_{34} = -42.0954400989517
x35=21.9875030069394x_{35} = 21.9875030069394
x36=82.3087554381567x_{36} = 82.3087554381567
x37=−99.9018455221845x_{37} = -99.9018455221845
x38=−55.91891815781x_{38} = -55.91891815781
x39=246.928858587175x_{39} = 246.928858587175
x40=−11.931302161923x_{40} = -11.931302161923
x41=−21.9875030069394x_{41} = -21.9875030069394
x42=−1.82420087382986x_{42} = -1.82420087382986
x43=−35.8119207078091x_{43} = -35.8119207078091
x44=92.3619577619375x_{44} = 92.3619577619375
x45=4.37901497732045x_{45} = 4.37901497732045
x46=−96.1319029246777x_{46} = -96.1319029246777
x47=48.3788725817987x_{47} = 48.3788725817987
x48=−15.7028490207672x_{48} = -15.7028490207672
x49=99.9018455221845x_{49} = 99.9018455221845
Puntos máximos de la función:
x49=43.9804772707051x_{49} = 43.9804772707051
x49=−43.9804772707051x_{49} = -43.9804772707051
x49=−47.7505322647089x_{49} = -47.7505322647089
x49=−27.6431177325273x_{49} = -27.6431177325273
x49=−10.0450569900272x_{49} = -10.0450569900272
x49=10.0450569900272x_{49} = 10.0450569900272
x49=77.9104708292441x_{49} = 77.9104708292441
x49=−77.9104708292441x_{49} = -77.9104708292441
x49=−56.5472525996449x_{49} = -56.5472525996449
x49=−54.0339126098678x_{49} = -54.0339126098678
x49=−52.7772402600661x_{49} = -52.7772402600661
x49=70.3705383836538x_{49} = 70.3705383836538
x49=−23.8727474905771x_{49} = -23.8727474905771
x49=−3.74702407259747x_{49} = -3.74702407259747
x49=−74.140507406587x_{49} = -74.140507406587
x49=−21.3590767768119x_{49} = -21.3590767768119
x49=−98.016874525578x_{49} = -98.016874525578
x49=−20.1022039761978x_{49} = -20.1022039761978
x49=84.1937327998049x_{49} = 84.1937327998049
x49=54.0339126098678x_{49} = 54.0339126098678
x49=64.0872415470646x_{49} = 64.0872415470646
x49=3.74702407259747x_{49} = 3.74702407259747
x49=26.3863422909371x_{49} = 26.3863422909371
x49=−81.680429427562x_{49} = -81.680429427562
x49=16.3313658042246x_{49} = 16.3313658042246
x49=23.8727474905771x_{49} = 23.8727474905771
x49=74.140507406587x_{49} = 74.140507406587
x49=−57.8039204446587x_{49} = -57.8039204446587
x49=40.2103952651705x_{49} = 40.2103952651705
x49=89.2203345987252x_{49} = 89.2203345987252
x49=−37.6969882430057x_{49} = -37.6969882430057
x49=6.27011200881177x_{49} = 6.27011200881177
x49=−13.8171889415972x_{49} = -13.8171889415972
x49=−8.78724240122631x_{49} = -8.78724240122631
x49=50.2638902610058x_{49} = 50.2638902610058
x49=30.1566338943204x_{49} = 30.1566338943204
x49=94.2469306832098x_{49} = 94.2469306832098
x49=−87.9636847230099x_{49} = -87.9636847230099
x49=−67.8572221291202x_{49} = -67.8572221291202
x49=87.9636847230099x_{49} = 87.9636847230099
x49=−33.9268407028905x_{49} = -33.9268407028905
x49=20.1022039761978x_{49} = 20.1022039761978
x49=98.016874525578x_{49} = 98.016874525578
x49=−91.7336332965327x_{49} = -91.7336332965327
x49=60.3172522833615x_{49} = 60.3172522833615
x49=−31.413377404844x_{49} = -31.413377404844
x49=−71.6271954015107x_{49} = -71.6271954015107
x49=−64.0872415470646x_{49} = -64.0872415470646
x49=33.9268407028905x_{49} = 33.9268407028905
Decrece en los intervalos
[246.928858587175,∞)\left[246.928858587175, \infty\right)
Crece en los intervalos
(−∞,−99.9018455221845]\left(-\infty, -99.9018455221845\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
20xsin⁡(5x)x2−1−25cos⁡(5x)+2(4x2x2−1−1)cos⁡(5x)x2−1x2−1=0\frac{\frac{20 x \sin{\left(5 x \right)}}{x^{2} - 1} - 25 \cos{\left(5 x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \cos{\left(5 x \right)}}{x^{2} - 1}}{x^{2} - 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−65.656848966939x_{1} = -65.656848966939
x2=34.2386827506632x_{2} = 34.2386827506632
x3=76.3386051877975x_{3} = 76.3386051877975
x4=38.0090585917967x_{4} = 38.0090585917967
x5=22.2981174970489x_{5} = 22.2981174970489
x6=−80.1086150586489x_{6} = -80.1086150586489
x7=68.1702129980947x_{7} = 68.1702129980947
x8=−27.9544434884427x_{8} = -27.9544434884427
x9=85.1352812841815x_{9} = 85.1352812841815
x10=36.1238828258751x_{10} = 36.1238828258751
x11=−123.463295267073x_{11} = -123.463295267073
x12=−7.8332076714592x_{12} = -7.8332076714592
x13=−49.9481185493933x_{13} = -49.9481185493933
x14=−51.8331907789564x_{14} = -51.8331907789564
x15=5.94128698602387x_{15} = 5.94128698602387
x16=−16.0120898954146x_{16} = -16.0120898954146
x17=4.04181360740579x_{17} = 4.04181360740579
x18=−43.6644716042339x_{18} = -43.6644716042339
x19=66.2851906144012x_{19} = 66.2851906144012
x20=−26.0690721965348x_{20} = -26.0690721965348
x21=−53.7182548129591x_{21} = -53.7182548129591
x22=−93.9319167827205x_{22} = -93.9319167827205
x23=−47.434674469425x_{23} = -47.434674469425
x24=−61.8867892164924x_{24} = -61.8867892164924
x25=26.0690721965348x_{25} = 26.0690721965348
x26=19.1553389817901x_{26} = 19.1553389817901
x27=78.2236113129099x_{27} = 78.2236113129099
x28=−54.9749600328382x_{28} = -54.9749600328382
x29=−50.5764769265287x_{29} = -50.5764769265287
x30=32.3534541001447x_{30} = 32.3534541001447
x31=100.215208922234x_{31} = 100.215208922234
x32=−14.1257815273595x_{32} = -14.1257815273595
x33=−4.04181360740579x_{33} = -4.04181360740579
x34=81.9936165892293x_{34} = 81.9936165892293
x35=−48.0630371582114x_{35} = -48.0630371582114
x36=24.183635752283x_{36} = 24.183635752283
x37=51.204834324493x_{37} = 51.204834324493
x38=−29.8397620458006x_{38} = -29.8397620458006
x39=−63.771821289198x_{39} = -63.771821289198
x40=71.9402472579385x_{40} = 71.9402472579385
x41=83.8786160540921x_{41} = 83.8786160540921
x42=−60.0017523341146x_{42} = -60.0017523341146
x43=46.1779454812218x_{43} = 46.1779454812218
x44=−39.8942135049353x_{44} = -39.8942135049353
x45=98.3302227144456x_{45} = 98.3302227144456
x46=75.7102692532699x_{46} = 75.7102692532699
x47=−38.0090585917967x_{47} = -38.0090585917967
x48=−31.7250373157656x_{48} = -31.7250373157656
x49=−97.7018937159058x_{49} = -97.7018937159058
x50=87.6486093200995x_{50} = 87.6486093200995
x51=88.2769408492032x_{51} = 88.2769408492032
x52=90.1619343490571x_{52} = 90.1619343490571
x53=−71.9402472579385x_{53} = -71.9402472579385
x54=10.3516496356449x_{54} = 10.3516496356449
x55=92.046926295711x_{55} = 92.046926295711
x56=65.0285068724037x_{56} = 65.0285068724037
x57=2.09906780363513x_{57} = 2.09906780363513
x58=−58.1167101734686x_{58} = -58.1167101734686
x59=60.0017523341146x_{59} = 60.0017523341146
x60=−75.7102692532699x_{60} = -75.7102692532699
x61=−68.7985529787465x_{61} = -68.7985529787465
x62=−19.7839250695181x_{62} = -19.7839250695181
x63=48.0630371582114x_{63} = 48.0630371582114
x64=12.2390481644834x_{64} = 12.2390481644834
x65=27.9544434884427x_{65} = 27.9544434884427
x66=−5.94128698602387x_{66} = -5.94128698602387
x67=0.400026104038616x_{67} = 0.400026104038616
x68=−41.7793503957282x_{68} = -41.7793503957282
x69=−85.7636135895293x_{69} = -85.7636135895293
x70=80.1086150586489x_{70} = 80.1086150586489
x71=39.8942135049353x_{71} = 39.8942135049353
x72=−87.6486093200995x_{72} = -87.6486093200995
x73=−9.72229932855872x_{73} = -9.72229932855872
x74=−73.8252596711376x_{74} = -73.8252596711376
x75=−2.09906780363513x_{75} = -2.09906780363513
x76=61.8867892164924x_{76} = 61.8867892164924
x77=−83.8786160540921x_{77} = -83.8786160540921
x78=49.9481185493933x_{78} = 49.9481185493933
x79=93.9319167827205x_{79} = 93.9319167827205
x80=−17.8981098836246x_{80} = -17.8981098836246
x81=−21.6695895084827x_{81} = -21.6695895084827
x82=−95.8169058962743x_{82} = -95.8169058962743
x83=16.0120898954146x_{83} = 16.0120898954146
x84=17.8981098836246x_{84} = 17.8981098836246
x85=14.1257815273595x_{85} = 14.1257815273595
x86=−92.6752566158296x_{86} = -92.6752566158296
x87=70.0552317849432x_{87} = 70.0552317849432
x88=−92.046926295711x_{88} = -92.046926295711
x89=57.4883615237795x_{89} = 57.4883615237795
x90=56.2316622030192x_{90} = 56.2316622030192
x91=54.3466078173532x_{91} = 54.3466078173532
x92=58.1167101734686x_{92} = 58.1167101734686
x93=−36.1238828258751x_{93} = -36.1238828258751
x94=44.2928422023494x_{94} = 44.2928422023494
x95=−70.0552317849432x_{95} = -70.0552317849432
x96=−81.9936165892293x_{96} = -81.9936165892293
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=−1x_{1} = -1
x2=1x_{2} = 1

lim⁡x→−1−(20xsin⁡(5x)x2−1−25cos⁡(5x)+2(4x2x2−1−1)cos⁡(5x)x2−1x2−1)=∞\lim_{x \to -1^-}\left(\frac{\frac{20 x \sin{\left(5 x \right)}}{x^{2} - 1} - 25 \cos{\left(5 x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \cos{\left(5 x \right)}}{x^{2} - 1}}{x^{2} - 1}\right) = \infty
lim⁡x→−1+(20xsin⁡(5x)x2−1−25cos⁡(5x)+2(4x2x2−1−1)cos⁡(5x)x2−1x2−1)=−∞\lim_{x \to -1^+}\left(\frac{\frac{20 x \sin{\left(5 x \right)}}{x^{2} - 1} - 25 \cos{\left(5 x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \cos{\left(5 x \right)}}{x^{2} - 1}}{x^{2} - 1}\right) = -\infty
- los límites no son iguales, signo
x1=−1x_{1} = -1
- es el punto de flexión
lim⁡x→1−(20xsin⁡(5x)x2−1−25cos⁡(5x)+2(4x2x2−1−1)cos⁡(5x)x2−1x2−1)=−∞\lim_{x \to 1^-}\left(\frac{\frac{20 x \sin{\left(5 x \right)}}{x^{2} - 1} - 25 \cos{\left(5 x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \cos{\left(5 x \right)}}{x^{2} - 1}}{x^{2} - 1}\right) = -\infty
lim⁡x→1+(20xsin⁡(5x)x2−1−25cos⁡(5x)+2(4x2x2−1−1)cos⁡(5x)x2−1x2−1)=∞\lim_{x \to 1^+}\left(\frac{\frac{20 x \sin{\left(5 x \right)}}{x^{2} - 1} - 25 \cos{\left(5 x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \cos{\left(5 x \right)}}{x^{2} - 1}}{x^{2} - 1}\right) = \infty
- los límites no son iguales, signo
x2=1x_{2} = 1
- es el punto de flexión

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[98.3302227144456,∞)\left[98.3302227144456, \infty\right)
Convexa en los intervalos
(−∞,−97.7018937159058]\left(-\infty, -97.7018937159058\right]
Asíntotas verticales
Hay:
x1=−1x_{1} = -1
x2=1x_{2} = 1
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(cos⁡(x+4x)x2−1)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x + 4 x \right)}}{x^{2} - 1}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(cos⁡(x+4x)x2−1)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x + 4 x \right)}}{x^{2} - 1}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(4*x + x)/(x^2 - 1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x+4x)x(x2−1))=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x + 4 x \right)}}{x \left(x^{2} - 1\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x+4x)x(x2−1))=0\lim_{x \to \infty}\left(\frac{\cos{\left(x + 4 x \right)}}{x \left(x^{2} - 1\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
cos⁡(x+4x)x2−1=cos⁡(5x)x2−1\frac{\cos{\left(x + 4 x \right)}}{x^{2} - 1} = \frac{\cos{\left(5 x \right)}}{x^{2} - 1}
- No
cos⁡(x+4x)x2−1=−cos⁡(5x)x2−1\frac{\cos{\left(x + 4 x \right)}}{x^{2} - 1} = - \frac{\cos{\left(5 x \right)}}{x^{2} - 1}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор