Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada$$- \frac{15 \pi x \sin{\left(\frac{\pi 15 x}{4} \right)}}{4} + \cos{\left(\frac{\pi 15 x}{4} \right)} + \frac{15 \pi \cos{\left(\frac{\pi 15 x}{4} \right)}}{4} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 94.1343115486088$$
$$x_{2} = 22.1374898155918$$
$$x_{3} = -88.0010463856655$$
$$x_{4} = -1.91050391760426$$
$$x_{5} = -45.8686739305448$$
$$x_{6} = 41.8688656060436$$
$$x_{7} = 80.267813852407$$
$$x_{8} = 18.1384042381761$$
$$x_{9} = 98.1342716797995$$
$$x_{10} = -9.87595386506925$$
$$x_{11} = -37.8690977390551$$
$$x_{12} = -57.8682578137898$$
$$x_{13} = -76.0012115786372$$
$$x_{14} = -56.0016441692461$$
$$x_{15} = -29.8697482874469$$
$$x_{16} = 72.5346028078849$$
$$x_{17} = -92.0010008959616$$
$$x_{18} = 54.1350341796472$$
$$x_{19} = -7.74514620610459$$
$$x_{20} = 50.1351698351176$$
$$x_{21} = 16.2723174671444$$
$$x_{22} = 31.7362338619054$$
$$x_{23} = 86.1344023931582$$
$$x_{24} = 66.1347256342552$$
$$x_{25} = 23.7372101014599$$
$$x_{26} = 34.1360300942816$$
$$x_{27} = -28.8031956241545$$
$$x_{28} = -85.8677390461176$$
$$x_{29} = 42.1355183629863$$
$$x_{30} = -53.8683759302086$$
$$x_{31} = -44.002092378808$$
$$x_{32} = -80.001151009115$$
$$x_{33} = 0.371963245781402$$
$$x_{34} = -48.001918090513$$
$$x_{35} = 74.1345754140356$$
$$x_{36} = 46.1353289998238$$
$$x_{37} = -96.000959196442$$
$$x_{38} = -23.2039657310033$$
$$x_{39} = -49.868512985502$$
$$x_{40} = -89.867691319953$$
$$x_{41} = -3.75719403358547$$
$$x_{42} = 52.2684282363304$$
$$x_{43} = -5.88214814157898$$
$$x_{44} = -72.0012788763162$$
$$x_{45} = 70.1346462415669$$
$$x_{46} = 82.1344544504577$$
$$x_{47} = -77.8678492053074$$
$$x_{48} = 6.14815881102693$$
$$x_{49} = -25.8702241841017$$
$$x_{50} = 58.1349171837128$$
$$x_{51} = -73.8679132306202$$
$$x_{52} = -64.0014387001558$$
$$x_{53} = -13.6067535068815$$
$$x_{54} = 8.27772837392236$$
$$x_{55} = 38.1357474165716$$
$$x_{56} = -41.8688656060436$$
$$x_{57} = -5.08451050564514$$
$$x_{58} = 26.1368546012308$$
$$x_{59} = 90.1343549554631$$
$$x_{60} = -65.8680646031785$$
$$x_{61} = 96.2676232063329$$
$$x_{62} = 62.1348152455554$$
$$x_{63} = 20.2712051205322$$
$$x_{64} = -21.8708737359272$$
$$x_{65} = -97.867607569677$$
$$x_{66} = 47.4686062995482$$
$$x_{67} = -81.8677914351114$$
$$x_{68} = -37.0691501716708$$
$$x_{69} = -17.8718130295761$$
$$x_{70} = 12.2741497925862$$
$$x_{71} = 10.1423784385639$$
$$x_{72} = -69.8679845850079$$
$$x_{73} = -32.0028763804354$$
$$x_{74} = -33.8693846441072$$
$$x_{75} = 37.0691501716708$$
$$x_{76} = 0.123696753065636$$
$$x_{77} = -100.000920832378$$
$$x_{78} = 68.8013383470347$$
$$x_{79} = -60.001534588503$$
$$x_{80} = 30.1363877123174$$
$$x_{81} = -61.8681549648286$$
$$x_{82} = -22.1374898155918$$
$$x_{83} = -93.8676476606577$$
$$x_{84} = -84.0010962068695$$
$$x_{85} = 24.270458372529$$
$$x_{86} = -13.8732909546253$$
$$x_{87} = -52.0017706001275$$
$$x_{88} = 4.02236183543391$$
$$x_{89} = 14.1398332426752$$
$$x_{90} = 72.801264825313$$
$$x_{91} = 2.17264396592635$$
$$x_{92} = 78.1345118366475$$
$$x_{93} = -40.0023014958459$$
$$x_{94} = -68.0013540890151$$
Signos de extremos en los puntos:
(94.13431154860879, 94.1343115486088*cos(1.00366830728296*pi) + sin(1.00366830728296*pi))
(22.137489815591838, 22.1374898155918*cos(1.01558680846939*pi) + sin(1.01558680846939*pi))
(-88.00104638566553, -sin(0.00392394624572034*pi) - 88.0010463856655*cos(0.00392394624572034*pi))
(-1.910503917604261, -sin(1.16438969101598*pi) - 1.91050391760426*cos(1.16438969101598*pi))
(-45.86867393054475, -sin(0.00752723954281009*pi) - 45.8686739305448*cos(0.00752723954281009*pi))
(41.868865606043634, 41.8688656060436*cos(1.00824602266363*pi) + sin(1.00824602266363*pi))
(80.267813852407, 80.267813852407*cos(1.00430194652625*pi) + sin(1.00430194652625*pi))
(18.138404238176086, 18.1384042381761*cos(0.0190158931603293*pi) + sin(0.0190158931603293*pi))
(98.13427167979953, 98.1342716797995*cos(0.00351879924824061*pi) + sin(0.00351879924824061*pi))
(-9.875953865069247, -sin(1.03482699400968*pi) - 9.87595386506925*cos(1.03482699400968*pi))
(-37.869097739055135, -sin(0.0091165214567468*pi) - 37.8690977390551*cos(0.0091165214567468*pi))
(-57.86825781378976, -sin(1.00596680171159*pi) - 57.8682578137898*cos(1.00596680171159*pi))
(-76.00121157863722, -sin(1.00454341988961*pi) - 76.0012115786372*cos(1.00454341988961*pi))
(-56.001644169246084, -sin(0.00616563467281139*pi) - 56.0016441692461*cos(0.00616563467281139*pi))
(-29.8697482874469, -sin(0.0115560779258743*pi) - 29.8697482874469*cos(0.0115560779258743*pi))
(72.53460280788485, 72.5346028078849*cos(0.00476052956821604*pi) + sin(0.00476052956821604*pi))
(-92.00100089596162, -sin(1.00375335985609*pi) - 92.0010008959616*cos(1.00375335985609*pi))
(54.135034179647185, 54.1350341796472*cos(1.00637817367695*pi) + sin(1.00637817367695*pi))
(-7.745146206104588, -sin(1.0442982728922*pi) - 7.74514620610459*cos(1.0442982728922*pi))
(50.13516983511756, 50.1351698351176*cos(0.00688688169083207*pi) + sin(0.00688688169083207*pi))
(16.27231746714436, 16.2723174671444*cos(1.02119050179135*pi) + sin(1.02119050179135*pi))
(31.736233861905358, 31.7362338619054*cos(1.01087698214509*pi) + sin(1.01087698214509*pi))
(86.13440239315817, 86.1344023931582*cos(1.00400897434315*pi) + sin(1.00400897434315*pi))
(66.13472563425522, 66.1347256342552*cos(0.00522112845706602*pi) + sin(0.00522112845706602*pi))
(23.73721010145992, 23.7372101014599*cos(1.01453788047469*pi) + sin(1.01453788047469*pi))
(34.136030094281615, 34.1360300942816*cos(0.010112853556052*pi) + sin(0.010112853556052*pi))
(-28.8031956241545, -sin(0.0119835905793764*pi) - 28.8031956241545*cos(0.0119835905793764*pi))
(-85.86773904611756, -sin(0.00402142294086616*pi) - 85.8677390461176*cos(0.00402142294086616*pi))
(42.13551836298631, 42.1355183629863*cos(0.00819386119866294*pi) + sin(0.00819386119866294*pi))
(-53.868375930208614, -sin(0.00640973828231495*pi) - 53.8683759302086*cos(0.00640973828231495*pi))
(-44.00209237880797, -sin(1.00784642052989*pi) - 44.002092378808*cos(1.00784642052989*pi))
(-80.00115100911502, -sin(0.00431628418130003*pi) - 80.001151009115*cos(0.00431628418130003*pi))
(0.37196324578140194, 0.371963245781402*cos(1.39486217168026*pi) + sin(1.39486217168026*pi))
(-48.00191809051297, -sin(0.00719283942365223*pi) - 48.001918090513*cos(0.00719283942365223*pi))
(74.13457541403561, 74.1345754140356*cos(0.00465780263351689*pi) + sin(0.00465780263351689*pi))
(46.13532899982377, 46.1353289998238*cos(1.00748374933914*pi) + sin(1.00748374933914*pi))
(-96.00095919644195, -sin(0.00359698665732822*pi) - 96.000959196442*cos(0.00359698665732822*pi))
(-23.20396573100327, -sin(1.01487149126226*pi) - 23.2039657310033*cos(1.01487149126226*pi))
(-49.868512985502, -sin(1.00692369563251*pi) - 49.868512985502*cos(1.00692369563251*pi))
(-89.86769131995298, -sin(1.00384244982371*pi) - 89.867691319953*cos(1.00384244982371*pi))
(-3.7571940335854737, -sin(0.089477625945527*pi) - 3.75719403358547*cos(0.089477625945527*pi))
(52.268428236330436, 52.2684282363304*cos(0.00660588623912872*pi) + sin(0.00660588623912872*pi))
(-5.882148141578975, -sin(0.0580555309211555*pi) - 5.88214814157898*cos(0.0580555309211555*pi))
(-72.00127887631623, -sin(0.00479578618586629*pi) - 72.0012788763162*cos(0.00479578618586629*pi))
(70.13464624156687, 70.1346462415669*cos(1.00492340587579*pi) + sin(1.00492340587579*pi))
(82.1344544504577, 82.1344544504577*cos(0.00420418921635246*pi) + sin(0.00420418921635246*pi))
(-77.86784920530744, -sin(0.00443451990287258*pi) - 77.8678492053074*cos(0.00443451990287258*pi))
(6.148158811026925, 6.14815881102693*cos(1.05559554135097*pi) + sin(1.05559554135097*pi))
(-25.870224184101694, -sin(1.01334069038136*pi) - 25.8702241841017*cos(1.01334069038136*pi))
(58.13491718371282, 58.1349171837128*cos(0.00593943892309312*pi) + sin(0.00593943892309312*pi))
(-73.86791323062023, -sin(1.00467461482589*pi) - 73.8679132306202*cos(1.00467461482589*pi))
(-64.00143870015577, -sin(0.00539512558415822*pi) - 64.0014387001558*cos(0.00539512558415822*pi))
(-13.606753506881475, -sin(1.02532565080553*pi) - 13.6067535068815*cos(1.02532565080553*pi))
(8.277728373922361, 8.27772837392236*cos(1.04148140220886*pi) + sin(1.04148140220886*pi))
(38.13574741657158, 38.1357474165716*cos(1.00905281214344*pi) + sin(1.00905281214344*pi))
(-41.868865606043634, -sin(1.00824602266363*pi) - 41.8688656060436*cos(1.00824602266363*pi))
(-5.084510505645143, -sin(1.06691439616929*pi) - 5.08451050564514*cos(1.06691439616929*pi))
(26.136854601230766, 26.1368546012308*cos(0.0132047546153728*pi) + sin(0.0132047546153728*pi))
(90.13435495546305, 90.1343549554631*cos(0.0038310829864372*pi) + sin(0.0038310829864372*pi))
(-65.86806460317854, -sin(1.00524226191953*pi) - 65.8680646031785*cos(1.00524226191953*pi))
(96.26762320633289, 96.2676232063329*cos(1.00358702374831*pi) + sin(1.00358702374831*pi))
(62.13481524555542, 62.1348152455554*cos(1.00555717083284*pi) + sin(1.00555717083284*pi))
(20.27120512053217, 20.2712051205322*cos(0.0170192019956374*pi) + sin(0.0170192019956374*pi))
(-21.870873735927184, -sin(0.0157765097269333*pi) - 21.8708737359272*cos(0.0157765097269333*pi))
(-97.86760756967698, -sin(1.00352838628868*pi) - 97.867607569677*cos(1.00352838628868*pi))
(47.46860629954816, 47.4686062995482*cos(0.00727362330562187*pi) + sin(0.00727362330562187*pi))
(-81.86779143511141, -sin(1.00421788166778*pi) - 81.8677914351114*cos(1.00421788166778*pi))
(-37.06915017167076, -sin(1.00931314376535*pi) - 37.0691501716708*cos(1.00931314376535*pi))
(-17.87181302957608, -sin(1.01929886091031*pi) - 17.8718130295761*cos(1.01929886091031*pi))
(12.274149792586227, 12.2741497925862*cos(0.0280617221983519*pi) + sin(0.0280617221983519*pi))
(10.142378438563917, 10.1423784385639*cos(0.0339191446146856*pi) + sin(0.0339191446146856*pi))
(-69.86798458500792, -sin(0.00494219377969785*pi) - 69.8679845850079*cos(0.00494219377969785*pi))
(-32.002876380435445, -sin(0.0107864266329187*pi) - 32.0028763804354*cos(0.0107864266329187*pi))
(-33.86938464410721, -sin(1.01019241540203*pi) - 33.8693846441072*cos(1.01019241540203*pi))
(37.06915017167076, 37.0691501716708*cos(1.00931314376535*pi) + sin(1.00931314376535*pi))
(0.12369675306563559, 0.123696753065636*cos(0.463862823996133*pi) + sin(0.463862823996133*pi))
(-100.00092083237837, -sin(1.00345312141889*pi) - 100.000920832378*cos(1.00345312141889*pi))
(68.8013383470347, 68.8013383470347*cos(0.00501880138011757*pi) + sin(0.00501880138011757*pi))
(-60.001534588503034, -sin(1.00575470688636*pi) - 60.001534588503*cos(1.00575470688636*pi))
(30.1363877123174, 30.1363877123174*cos(1.01145392119025*pi) + sin(1.01145392119025*pi))
(-61.86815496482863, -sin(0.00558111810738637*pi) - 61.8681549648286*cos(0.00558111810738637*pi))
(-22.137489815591838, -sin(1.01558680846939*pi) - 22.1374898155918*cos(1.01558680846939*pi))
(-93.8676476606577, -sin(0.00367872746636522*pi) - 93.8676476606577*cos(0.00367872746636522*pi))
(-84.00109620686948, -sin(1.00411077576052*pi) - 84.0010962068695*cos(1.00411077576052*pi))
(24.27045837252896, 24.270458372529*cos(1.01421889698361*pi) + sin(1.01421889698361*pi))
(-13.87329095462527, -sin(0.0248410798447622*pi) - 13.8732909546253*cos(0.0248410798447622*pi))
(-52.001770600127486, -sin(1.00663975047809*pi) - 52.0017706001275*cos(1.00663975047809*pi))
(4.022361835433905, 4.02236183543391*cos(1.08385688287714*pi) + sin(1.08385688287714*pi))
(14.139833242675245, 14.1398332426752*cos(1.02437466003217*pi) + sin(1.02437466003217*pi))
(72.80126482531298, 72.801264825313*cos(1.0047430949237*pi) + sin(1.0047430949237*pi))
(2.172643965926345, 2.17264396592635*cos(0.147414872223793*pi) + sin(0.147414872223793*pi))
(78.1345118366475, 78.1345118366475*cos(1.00441938742813*pi) + sin(1.00441938742813*pi))
(-40.002301495845884, -sin(0.00863060942205607*pi) - 40.0023014958459*cos(0.00863060942205607*pi))
(-68.0013540890151, -sin(1.00507783380661*pi) - 68.0013540890151*cos(1.00507783380661*pi))
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:
$$x_{1} = 94.1343115486088$$
$$x_{2} = 22.1374898155918$$
$$x_{3} = -88.0010463856655$$
$$x_{4} = -45.8686739305448$$
$$x_{5} = 41.8688656060436$$
$$x_{6} = 80.267813852407$$
$$x_{7} = -37.8690977390551$$
$$x_{8} = -56.0016441692461$$
$$x_{9} = -29.8697482874469$$
$$x_{10} = 54.1350341796472$$
$$x_{11} = 16.2723174671444$$
$$x_{12} = 31.7362338619054$$
$$x_{13} = 86.1344023931582$$
$$x_{14} = 23.7372101014599$$
$$x_{15} = -28.8031956241545$$
$$x_{16} = -85.8677390461176$$
$$x_{17} = -53.8683759302086$$
$$x_{18} = -80.001151009115$$
$$x_{19} = 0.371963245781402$$
$$x_{20} = -48.001918090513$$
$$x_{21} = 46.1353289998238$$
$$x_{22} = -96.000959196442$$
$$x_{23} = -3.75719403358547$$
$$x_{24} = -5.88214814157898$$
$$x_{25} = -72.0012788763162$$
$$x_{26} = 70.1346462415669$$
$$x_{27} = -77.8678492053074$$
$$x_{28} = 6.14815881102693$$
$$x_{29} = -64.0014387001558$$
$$x_{30} = 8.27772837392236$$
$$x_{31} = 38.1357474165716$$
$$x_{32} = 96.2676232063329$$
$$x_{33} = 62.1348152455554$$
$$x_{34} = -21.8708737359272$$
$$x_{35} = -69.8679845850079$$
$$x_{36} = -32.0028763804354$$
$$x_{37} = 37.0691501716708$$
$$x_{38} = 30.1363877123174$$
$$x_{39} = -61.8681549648286$$
$$x_{40} = -93.8676476606577$$
$$x_{41} = 24.270458372529$$
$$x_{42} = -13.8732909546253$$
$$x_{43} = 4.02236183543391$$
$$x_{44} = 14.1398332426752$$
$$x_{45} = 72.801264825313$$
$$x_{46} = 78.1345118366475$$
$$x_{47} = -40.0023014958459$$
Puntos máximos de la función:
$$x_{47} = -1.91050391760426$$
$$x_{47} = 18.1384042381761$$
$$x_{47} = 98.1342716797995$$
$$x_{47} = -9.87595386506925$$
$$x_{47} = -57.8682578137898$$
$$x_{47} = -76.0012115786372$$
$$x_{47} = 72.5346028078849$$
$$x_{47} = -92.0010008959616$$
$$x_{47} = -7.74514620610459$$
$$x_{47} = 50.1351698351176$$
$$x_{47} = 66.1347256342552$$
$$x_{47} = 34.1360300942816$$
$$x_{47} = 42.1355183629863$$
$$x_{47} = -44.002092378808$$
$$x_{47} = 74.1345754140356$$
$$x_{47} = -23.2039657310033$$
$$x_{47} = -49.868512985502$$
$$x_{47} = -89.867691319953$$
$$x_{47} = 52.2684282363304$$
$$x_{47} = 82.1344544504577$$
$$x_{47} = -25.8702241841017$$
$$x_{47} = 58.1349171837128$$
$$x_{47} = -73.8679132306202$$
$$x_{47} = -13.6067535068815$$
$$x_{47} = -41.8688656060436$$
$$x_{47} = -5.08451050564514$$
$$x_{47} = 26.1368546012308$$
$$x_{47} = 90.1343549554631$$
$$x_{47} = -65.8680646031785$$
$$x_{47} = 20.2712051205322$$
$$x_{47} = -97.867607569677$$
$$x_{47} = 47.4686062995482$$
$$x_{47} = -81.8677914351114$$
$$x_{47} = -37.0691501716708$$
$$x_{47} = -17.8718130295761$$
$$x_{47} = 12.2741497925862$$
$$x_{47} = 10.1423784385639$$
$$x_{47} = -33.8693846441072$$
$$x_{47} = 0.123696753065636$$
$$x_{47} = -100.000920832378$$
$$x_{47} = 68.8013383470347$$
$$x_{47} = -60.001534588503$$
$$x_{47} = -22.1374898155918$$
$$x_{47} = -84.0010962068695$$
$$x_{47} = -52.0017706001275$$
$$x_{47} = 2.17264396592635$$
$$x_{47} = -68.0013540890151$$
Decrece en los intervalos
$$\left[96.2676232063329, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -96.000959196442\right]$$