Gráfico de la función y = cos(3*π*x)/x

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
$$- \frac{3 \pi \sin{\left(3 \pi x \right)}}{x} - \frac{\cos{\left(3 \pi x \right)}}{x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -61.9998184204633$$
$$x_{2} = -97.9998851232848$$
$$x_{3} = -75.9998518694221$$
$$x_{4} = 91.9998776313122$$
$$x_{5} = -7.99859259627383$$
$$x_{6} = -67.9998344422417$$
$$x_{7} = -57.9998058976826$$
$$x_{8} = 75.9998518694221$$
$$x_{9} = 51.9997835011434$$
$$x_{10} = -41.9997319534002$$
$$x_{11} = 57.9998058976826$$
$$x_{12} = 45.9997552619734$$
$$x_{13} = -37.9997037376893$$
$$x_{14} = -29.9996247332274$$
$$x_{15} = -65.9998294253229$$
$$x_{16} = -19.9994370939732$$
$$x_{17} = -47.999765459459$$
$$x_{18} = 23.9995309143338$$
$$x_{19} = 93.9998802349058$$
$$x_{20} = 71.9998436399223$$
$$x_{21} = -73.9998478658821$$
$$x_{22} = -45.9997552619734$$
$$x_{23} = -91.9998776313122$$
$$x_{24} = -79.9998592759688$$
$$x_{25} = 95.999882730016$$
$$x_{26} = -77.9998556676515$$
$$x_{27} = 97.9998851232848$$
$$x_{28} = 59.9998123677873$$
$$x_{29} = 67.9998344422417$$
$$x_{30} = -83.9998659771278$$
$$x_{31} = -85.9998690939451$$
$$x_{32} = 77.9998556676515$$
$$x_{33} = -99.9998874208226$$
$$x_{34} = 11.9990617919904$$
$$x_{35} = 79.9998592759688$$
$$x_{36} = -81.99986270827$$
$$x_{37} = -11.9990617919904$$
$$x_{38} = -39.9997185509474$$
$$x_{39} = 1.99436044482429$$
$$x_{40} = 39.9997185509474$$
$$x_{41} = 81.99986270827$$
$$x_{42} = 49.9997748411382$$
$$x_{43} = 87.9998720690886$$
$$x_{44} = 5.99812329045085$$
$$x_{45} = 99.9998874208226$$
$$x_{46} = -23.9995309143338$$
$$x_{47} = -1.99436044482429$$
$$x_{48} = -33.999668882871$$
$$x_{49} = -13.9991958328274$$
$$x_{50} = -17.9993745461062$$
$$x_{51} = 55.9997989654243$$
$$x_{52} = -87.9998720690886$$
$$x_{53} = 69.999839172478$$
$$x_{54} = 33.999668882871$$
$$x_{55} = -59.9998123677873$$
$$x_{56} = 19.9994370939732$$
$$x_{57} = -93.9998802349058$$
$$x_{58} = 21.9994882689149$$
$$x_{59} = 31.999648187756$$
$$x_{60} = 9.99887412456459$$
$$x_{61} = -71.9998436399223$$
$$x_{62} = -55.9997989654243$$
$$x_{63} = -21.9994882689149$$
$$x_{64} = 65.9998294253229$$
$$x_{65} = 15.9992963600396$$
$$x_{66} = 61.9998184204633$$
$$x_{67} = 3.99718420125386$$
$$x_{68} = 37.9997037376893$$
$$x_{69} = 73.9998478658821$$
$$x_{70} = -5.99812329045085$$
$$x_{71} = 25.9995669986813$$
$$x_{72} = -51.9997835011434$$
$$x_{73} = 41.9997319534002$$
$$x_{74} = 27.9995979279619$$
$$x_{75} = -53.9997915196617$$
$$x_{76} = 53.9997915196617$$
$$x_{77} = 17.9993745461062$$
$$x_{78} = 83.9998659771278$$
$$x_{79} = 7.99859259627383$$
$$x_{80} = -9.99887412456459$$
$$x_{81} = -89.9998749120031$$
$$x_{82} = -49.9997748411382$$
$$x_{83} = -31.999648187756$$
$$x_{84} = -15.9992963600396$$
$$x_{85} = -3.99718420125386$$
$$x_{86} = -63.999824094845$$
$$x_{87} = -95.999882730016$$
$$x_{88} = 89.9998749120031$$
$$x_{89} = 63.999824094845$$
$$x_{90} = 29.9996247332274$$
$$x_{91} = 13.9991958328274$$
$$x_{92} = 85.9998690939451$$
$$x_{93} = -35.9996872784864$$
$$x_{94} = 47.999765459459$$
$$x_{95} = -25.9995669986813$$
$$x_{96} = 35.9996872784864$$
$$x_{97} = 43.9997441374332$$
$$x_{98} = -69.999839172478$$
$$x_{99} = -27.9995979279619$$
$$x_{100} = -43.9997441374332$$
Signos de extremos en los puntos:

(-61.99981842046332, -0.0161290794953352*cos(1.99945526138995*pi))

(-97.99988512328478, -0.0102040935940077*cos(1.99965536985434*pi))

(-75.99985186942207, -0.0131579203827678*cos(1.99955560826623*pi))

(91.99987763131222, 0.0108695796749587*cos(1.99963289393668*pi))

(-7.998592596273832, -0.125021994552623*cos(1.9957777888215*pi))

(-67.99983444224166, -0.0147059181570418*cos(1.99950332672501*pi))

(-57.99980589768257, -0.0172414370103945*cos(1.99941769304772*pi))

(75.99985186942207, 0.0131579203827678*cos(1.99955560826623*pi))

(51.99978350114343, 0.0192308492972478*cos(1.99935050343029*pi))

(-41.99973195340017, -0.0238096757643484*cos(1.99919586020052*pi))

(57.99980589768257, 0.0172414370103945*cos(1.99941769304772*pi))

(45.99975526197338, 0.0217392460960911*cos(1.99926578592013*pi))

(-37.999703737689266, -0.026315994643089*cos(1.99911121306781*pi))

(-29.999624733227435, -0.0333337503016298*cos(1.99887419968231*pi))

(-65.99982942532287, -0.015151554310174*cos(1.99948827596862*pi))

(-19.999437093973224, -0.050001407304676*cos(1.99831128191967*pi))

(-47.999765459459034, -0.0208334351309405*cos(1.9992963783771*pi))

(23.999530914333842, 0.0416674810674214*cos(1.99859274300152*pi))

(93.99988023490579, 0.0106383114265784*cos(1.99964070471736*pi))

(71.99984363992233, 0.0138889190510064*cos(1.99953091976698*pi))

(-73.99984786588212, -0.0135135412955498*cos(1.99954359764635*pi))

(-45.99975526197338, -0.0217392460960911*cos(1.99926578592013*pi))

(-91.99987763131222, -0.0108695796749587*cos(1.99963289393668*pi))

(-79.9998592759688, -0.0125000219881686*cos(1.99957782790642*pi))

(95.99988273001603, 0.0104166793912898*cos(1.99964819004811*pi))

(-77.99985566765152, -0.0128205365438224*cos(1.99956700295456*pi))

(97.99988512328478, 0.0102040935940077*cos(1.99965536985434*pi))

(59.999812367787264, 0.0166667187868887*cos(1.9994371033618*pi))

(67.99983444224166, 0.0147059181570418*cos(1.99950332672501*pi))

(-83.99986597712775, -0.0119047808989634*cos(1.99959793138328*pi))

(-85.99986909394514, -0.0116279246763459*cos(1.99960728183544*pi))

(77.99985566765152, 0.0128205365438224*cos(1.99956700295456*pi))

(-99.99988742082257, -0.0100000112579304*cos(1.99966226246772*pi))

(11.999061791990425, 0.0833398491761678*cos(1.99718537597127*pi))

(79.9998592759688, 0.0125000219881686*cos(1.99957782790642*pi))

(-81.99986270827, -0.012195142369417*cos(1.99958812481*pi))

(-11.999061791990425, -0.0833398491761678*cos(1.99718537597127*pi))

(-39.999718550947435, -0.0250001759068956*cos(1.9991556528423*pi))

(1.994360444824287, 0.501413875608682*cos(1.98308133447286*pi))

(39.999718550947435, 0.0250001759068956*cos(1.9991556528423*pi))

(81.99986270827, 0.012195142369417*cos(1.99958812481*pi))

(49.999774841138176, 0.0200000900639503*cos(1.99932452341454*pi))

(87.99987206908858, 0.0113636528836644*cos(1.99961620726572*pi))

(5.998123290450845, 0.166718813798313*cos(1.99436987135254*pi))

(99.99988742082257, 0.0100000112579304*cos(1.99966226246772*pi))

(-23.999530914333842, -0.0416674810674214*cos(1.99859274300152*pi))

(-1.994360444824287, -0.501413875608682*cos(1.98308133447286*pi))

(-33.999668882871, -0.0294120511421745*cos(1.999006648613*pi))

(-13.999195832827429, -0.0714326745579949*cos(1.99758749848229*pi))

(-17.999374546106203, -0.0555574860358873*cos(1.99812363831861*pi))

(55.99979896542434, 0.0178572069627861*cos(1.99939689627303*pi))

(-87.99987206908858, -0.0113636528836644*cos(1.99961620726572*pi))

(69.99983917247803, 0.014285747107733*cos(1.9995175174341*pi))

(33.999668882871, 0.0294120511421745*cos(1.999006648613*pi))

(-59.999812367787264, -0.0166667187868887*cos(1.9994371033618*pi))

(19.999437093973224, 0.050001407304676*cos(1.99831128191967*pi))

(-93.99988023490579, -0.0106383114265784*cos(1.99964070471736*pi))

(21.99948826891488, 0.0454556027747697*cos(1.99846480674464*pi))

(31.999648187755994, 0.0312503435704218*cos(1.99894456326798*pi))

(9.998874124564592, 0.100011260022092*cos(1.99662237369377*pi))

(-71.99984363992233, -0.0138889190510064*cos(1.99953091976698*pi))

(-55.99979896542434, -0.0178572069627861*cos(1.99939689627303*pi))

(-21.99948826891488, -0.0454556027747697*cos(1.99846480674464*pi))

(65.99982942532287, 0.015151554310174*cos(1.99948827596862*pi))

(15.999296360039631, 0.0625027487144768*cos(1.9978890801189*pi))

(61.99981842046332, 0.0161290794953352*cos(1.99945526138995*pi))

(3.997184201253863, 0.250176111395195*cos(1.99155260376159*pi))

(37.999703737689266, 0.026315994643089*cos(1.99911121306781*pi))

(73.99984786588212, 0.0135135412955498*cos(1.99954359764635*pi))

(-5.998123290450845, -0.166718813798313*cos(1.99436987135254*pi))

(25.99956699868127, 0.0384621790067012*cos(1.99870099604381*pi))

(-51.99978350114343, -0.0192308492972478*cos(1.99935050343029*pi))

(41.99973195340017, 0.0238096757643484*cos(1.99919586020052*pi))

(27.99959792796185, 0.0357147985686376*cos(1.99879378388556*pi))

(-53.99979151966167, -0.0185185900141095*cos(1.99937455898501*pi))

(53.99979151966167, 0.0185185900141095*cos(1.99937455898501*pi))

(17.999374546106203, 0.0555574860358873*cos(1.99812363831861*pi))

(83.99986597712775, 0.0119047808989634*cos(1.99959793138328*pi))

(7.998592596273832, 0.125021994552623*cos(1.9957777888215*pi))

(-9.998874124564592, -0.100011260022092*cos(1.99662237369377*pi))

(-89.99987491200305, -0.0111111265540952*cos(1.99962473600914*pi))

(-49.999774841138176, -0.0200000900639503*cos(1.99932452341454*pi))

(-31.999648187755994, -0.0312503435704218*cos(1.99894456326798*pi))

(-15.999296360039631, -0.0625027487144768*cos(1.9978890801189*pi))

(-3.997184201253863, -0.250176111395195*cos(1.99155260376159*pi))

(-63.99982409484497, -0.0156250429457125*cos(1.9994722845349*pi))

(-95.99988273001603, -0.0104166793912898*cos(1.99964819004811*pi))

(89.99987491200305, 0.0111111265540952*cos(1.99962473600914*pi))

(63.99982409484497, 0.0156250429457125*cos(1.9994722845349*pi))

(29.999624733227435, 0.0333337503016298*cos(1.99887419968231*pi))

(13.999195832827429, 0.0714326745579949*cos(1.99758749848229*pi))

(85.99986909394514, 0.0116279246763459*cos(1.99960728183544*pi))

(-35.999687278486384, -0.0277780190773381*cos(1.99906183545914*pi))

(47.999765459459034, 0.0208334351309405*cos(1.9992963783771*pi))

(-25.99956699868127, -0.0384621790067012*cos(1.99870099604381*pi))

(35.999687278486384, 0.0277780190773381*cos(1.99906183545914*pi))

(43.99974413743324, 0.022727404888458*cos(1.9992324122997*pi))

(-69.99983917247803, -0.014285747107733*cos(1.9995175174341*pi))

(-27.99959792796185, -0.0357147985686376*cos(1.99879378388556*pi))

(-43.99974413743324, -0.022727404888458*cos(1.9992324122997*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:
$$x_{1} = -61.9998184204633$$
$$x_{2} = -97.9998851232848$$
$$x_{3} = -75.9998518694221$$
$$x_{4} = -7.99859259627383$$
$$x_{5} = -67.9998344422417$$
$$x_{6} = -57.9998058976826$$
$$x_{7} = -41.9997319534002$$
$$x_{8} = -37.9997037376893$$
$$x_{9} = -29.9996247332274$$
$$x_{10} = -65.9998294253229$$
$$x_{11} = -19.9994370939732$$
$$x_{12} = -47.999765459459$$
$$x_{13} = -73.9998478658821$$
$$x_{14} = -45.9997552619734$$
$$x_{15} = -91.9998776313122$$
$$x_{16} = -79.9998592759688$$
$$x_{17} = -77.9998556676515$$
$$x_{18} = -83.9998659771278$$
$$x_{19} = -85.9998690939451$$
$$x_{20} = -99.9998874208226$$
$$x_{21} = -81.99986270827$$
$$x_{22} = -11.9990617919904$$
$$x_{23} = -39.9997185509474$$
$$x_{24} = -23.9995309143338$$
$$x_{25} = -1.99436044482429$$
$$x_{26} = -33.999668882871$$
$$x_{27} = -13.9991958328274$$
$$x_{28} = -17.9993745461062$$
$$x_{29} = -87.9998720690886$$
$$x_{30} = -59.9998123677873$$
$$x_{31} = -93.9998802349058$$
$$x_{32} = -71.9998436399223$$
$$x_{33} = -55.9997989654243$$
$$x_{34} = -21.9994882689149$$
$$x_{35} = -5.99812329045085$$
$$x_{36} = -51.9997835011434$$
$$x_{37} = -53.9997915196617$$
$$x_{38} = -9.99887412456459$$
$$x_{39} = -89.9998749120031$$
$$x_{40} = -49.9997748411382$$
$$x_{41} = -31.999648187756$$
$$x_{42} = -15.9992963600396$$
$$x_{43} = -3.99718420125386$$
$$x_{44} = -63.999824094845$$
$$x_{45} = -95.999882730016$$
$$x_{46} = -35.9996872784864$$
$$x_{47} = -25.9995669986813$$
$$x_{48} = -69.999839172478$$
$$x_{49} = -27.9995979279619$$
$$x_{50} = -43.9997441374332$$
Puntos máximos de la función:
$$x_{50} = 91.9998776313122$$
$$x_{50} = 75.9998518694221$$
$$x_{50} = 51.9997835011434$$
$$x_{50} = 57.9998058976826$$
$$x_{50} = 45.9997552619734$$
$$x_{50} = 23.9995309143338$$
$$x_{50} = 93.9998802349058$$
$$x_{50} = 71.9998436399223$$
$$x_{50} = 95.999882730016$$
$$x_{50} = 97.9998851232848$$
$$x_{50} = 59.9998123677873$$
$$x_{50} = 67.9998344422417$$
$$x_{50} = 77.9998556676515$$
$$x_{50} = 11.9990617919904$$
$$x_{50} = 79.9998592759688$$
$$x_{50} = 1.99436044482429$$
$$x_{50} = 39.9997185509474$$
$$x_{50} = 81.99986270827$$
$$x_{50} = 49.9997748411382$$
$$x_{50} = 87.9998720690886$$
$$x_{50} = 5.99812329045085$$
$$x_{50} = 99.9998874208226$$
$$x_{50} = 55.9997989654243$$
$$x_{50} = 69.999839172478$$
$$x_{50} = 33.999668882871$$
$$x_{50} = 19.9994370939732$$
$$x_{50} = 21.9994882689149$$
$$x_{50} = 31.999648187756$$
$$x_{50} = 9.99887412456459$$
$$x_{50} = 65.9998294253229$$
$$x_{50} = 15.9992963600396$$
$$x_{50} = 61.9998184204633$$
$$x_{50} = 3.99718420125386$$
$$x_{50} = 37.9997037376893$$
$$x_{50} = 73.9998478658821$$
$$x_{50} = 25.9995669986813$$
$$x_{50} = 41.9997319534002$$
$$x_{50} = 27.9995979279619$$
$$x_{50} = 53.9997915196617$$
$$x_{50} = 17.9993745461062$$
$$x_{50} = 83.9998659771278$$
$$x_{50} = 7.99859259627383$$
$$x_{50} = 89.9998749120031$$
$$x_{50} = 63.999824094845$$
$$x_{50} = 29.9996247332274$$
$$x_{50} = 13.9991958328274$$
$$x_{50} = 85.9998690939451$$
$$x_{50} = 47.999765459459$$
$$x_{50} = 35.9996872784864$$
$$x_{50} = 43.9997441374332$$
Decrece en los intervalos
$$\left[-1.99436044482429, 1.99436044482429\right]$$
Crece en los intervalos
$$\left(-\infty, -99.9998874208226\right]$$