Gráfico de la función y = cos(pi*x)/(x-2)

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
$$- \frac{\pi \sin{\left(\pi x \right)}}{x - 2} - \frac{\cos{\left(\pi x \right)}}{\left(x - 2\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 87.9988218359416$$
$$x_{2} = -53.9981906541763$$
$$x_{3} = -11.9927602767546$$
$$x_{4} = 31.9966223736938$$
$$x_{5} = 15.9927602767546$$
$$x_{6} = -81.9987937838856$$
$$x_{7} = -85.99884861287$$
$$x_{8} = -17.9949330850375$$
$$x_{9} = 35.9970197910376$$
$$x_{10} = 19.9943698713525$$
$$x_{11} = -9.99155260376159$$
$$x_{12} = 29.9963810744755$$
$$x_{13} = -49.9980514670226$$
$$x_{14} = 81.998733471837$$
$$x_{15} = 0.0515454605792004$$
$$x_{16} = -43.9977972952556$$
$$x_{17} = 79.9987009960438$$
$$x_{18} = -91.9989221068275$$
$$x_{19} = 59.998253047962$$
$$x_{20} = 89.99884861287$$
$$x_{21} = -77.998733471837$$
$$x_{22} = -83.9988218359416$$
$$x_{23} = -21.9957777888215$$
$$x_{24} = 69.9985099608267$$
$$x_{25} = -99.999006648613$$
$$x_{26} = 33.9968335041224$$
$$x_{27} = 97.998944563268$$
$$x_{28} = 93.9988986739972$$
$$x_{29} = 41.9974668634628$$
$$x_{30} = 23.9953938487289$$
$$x_{31} = 39.9973335283242$$
$$x_{32} = -15.9943698713525$$
$$x_{33} = -41.9976971654773$$
$$x_{34} = -67.9985525345655$$
$$x_{35} = 5.97456187738277$$
$$x_{36} = 75.9986307779258$$
$$x_{37} = -89.9988986739972$$
$$x_{38} = -33.9971853759713$$
$$x_{39} = 57.9981906541763$$
$$x_{40} = -93.998944563268$$
$$x_{41} = 37.9971853759713$$
$$x_{42} = 21.9949330850375$$
$$x_{43} = 95.9989221068275$$
$$x_{44} = 91.9988741996823$$
$$x_{45} = -97.9989867813195$$
$$x_{46} = -3.98308133447286$$
$$x_{47} = -61.998416830397$$
$$x_{48} = -79.9987643633963$$
$$x_{49} = 63.9983657586429$$
$$x_{50} = -55.998253047962$$
$$x_{51} = -39.9975874984823$$
$$x_{52} = -13.9936657542665$$
$$x_{53} = -45.9978890801189$$
$$x_{54} = -73.9986668109399$$
$$x_{55} = 71.9985525345655$$
$$x_{56} = -59.9983657586429$$
$$x_{57} = 53.9980514670226$$
$$x_{58} = -71.9986307779258$$
$$x_{59} = 17.9936657542665$$
$$x_{60} = -65.9985099608267$$
$$x_{61} = -23.9961026419285$$
$$x_{62} = -31.9970197910376$$
$$x_{63} = -25.9963810744755$$
$$x_{64} = -7.98986102861818$$
$$x_{65} = 7.98308133447286$$
$$x_{66} = 43.9975874984823$$
$$x_{67} = -37.9974668634628$$
$$x_{68} = -69.9985927430015$$
$$x_{69} = 11.9898610286182$$
$$x_{70} = -35.9973335283242$$
$$x_{71} = -27.9966223736938$$
$$x_{72} = 77.9986668109399$$
$$x_{73} = -1.97456187738277$$
$$x_{74} = -19.9953938487289$$
$$x_{75} = 67.9984648067446$$
$$x_{76} = -5.98732145729946$$
$$x_{77} = -87.9988741996823$$
$$x_{78} = 65.998416830397$$
$$x_{79} = -63.9984648067446$$
$$x_{80} = -51.9981236383186$$
$$x_{81} = 85.9987937838856$$
$$x_{82} = 45.9976971654773$$
$$x_{83} = 49.9978890801189$$
$$x_{84} = -95.9989661030993$$
$$x_{85} = -47.9979735215724$$
$$x_{86} = 25.9957777888215$$
$$x_{87} = 99.9989661030993$$
$$x_{88} = -29.9968335041224$$
$$x_{89} = 27.9961026419285$$
$$x_{90} = 83.9987643633963$$
$$x_{91} = 61.9983112819197$$
$$x_{92} = 51.9979735215724$$
$$x_{93} = -57.9983112819197$$
$$x_{94} = 3.9484545394208$$
$$x_{95} = 55.9981236383186$$
$$x_{96} = 73.9985927430015$$
$$x_{97} = 9.98732145729946$$
$$x_{98} = -75.9987009960438$$
$$x_{99} = 13.9915526037616$$
$$x_{100} = 47.9977972952556$$
Signos de extremos en los puntos:

(87.9988218359416, 0.0116280662763925*cos(1.9988218359416*pi))

(-53.998190654176284, -0.0178577198355501*cos(1.99819065417628*pi))

(-11.99276027675465, -0.0714655279031144*cos(1.99276027675465*pi))

(31.996622373693775, 0.0333370866740308*cos(1.99662237369377*pi))

(15.99276027675465, 0.0714655279031144*cos(1.99276027675465*pi))

(-81.99879378388556, -0.0119049328562125*cos(1.99879378388556*pi))

(-85.99884861287003, -0.011363785046771*cos(1.99884861287003*pi))

(-17.994933085037538, -0.0500126704974228*cos(1.99493308503754*pi))

(35.99701979103759, 0.0294143429673098*cos(1.99701979103759*pi))

(19.994369871352536, 0.0555729379327711*cos(1.99436987135254*pi))

(-9.991552603761589, -0.0833920371317317*cos(1.99155260376159*pi))

(29.9963810744755, 0.0357189022873998*cos(1.9963810744755*pi))

(-49.99805146702257, -0.0192314898690811*cos(1.99805146702257*pi))

(81.99873347183703, 0.0125001978981585*cos(1.99873347183703*pi))

(0.05154546057920037, -0.513227267954253*cos(0.0515454605792004*pi))

(-43.997797295255594, -0.0217401714604091*cos(1.99779729525559*pi))

(79.99870099604381, 0.0128207263355671*cos(1.99870099604381*pi))

(-91.99892210682754, -0.0106384198625546*cos(1.99892210682754*pi))

(59.998253047962045, 0.0172418986339648*cos(1.99825304796205*pi))

(89.99884861287003, 0.011363785046771*cos(1.99884861287003*pi))

(-77.99873347183703, -0.0125001978981585*cos(1.99873347183703*pi))

(-83.9988218359416, -0.0116280662763925*cos(1.9988218359416*pi))

(-21.995777788821496, -0.0416739981842078*cos(1.9957777888215*pi))

(69.99850996082668, 0.014706204600308*cos(1.99850996082668*pi))

(-99.999006648613, -0.00980401704739149*cos(1.999006648613*pi))

(33.996833504122364, 0.0312530925871513*cos(1.99683350412236*pi))

(97.99894456326798, 0.0104167811901406*cos(1.99894456326798*pi))

(93.99889867399723, 0.0108696953378056*cos(1.99889867399723*pi))

(41.99746686346282, 0.0250015833106043*cos(1.99746686346282*pi))

(23.99539384872894, 0.0454640642889778*cos(1.99539384872894*pi))

(39.99733352832425, 0.026317636190302*cos(1.99733352832425*pi))

(-15.994369871352536, -0.0555729379327711*cos(1.99436987135254*pi))

(-41.997697165477256, -0.0227284622701719*cos(1.99769716547726*pi))

(-67.99855253456555, -0.0142860096929318*cos(1.99855253456555*pi))

(5.974561877382775, 0.251600058283278*cos(1.97456187738277*pi))

(75.99863077792584, 0.0135137635586942*cos(1.99863077792584*pi))

(-89.99889867399723, -0.0108696953378056*cos(1.99889867399723*pi))

(-33.99718537597127, -0.0277799497253893*cos(1.99718537597127*pi))

(57.998190654176284, 0.0178577198355501*cos(1.99819065417628*pi))

(-93.99894456326798, -0.0104167811901406*cos(1.99894456326798*pi))

(37.99718537597127, 0.0277799497253893*cos(1.99718537597127*pi))

(21.994933085037538, 0.0500126704974228*cos(1.99493308503754*pi))

(95.99892210682754, 0.0106384198625546*cos(1.99892210682754*pi))

(91.9988741996823, 0.0111112501005433*cos(1.99887419968231*pi))

(-97.9989867813195, -0.0100001013228947*cos(1.9989867813195*pi))

(-3.9830813344728613, -0.167137958536227*cos(1.98308133447286*pi))

(-61.998416830397005, -0.0156253865255779*cos(1.99841683039701*pi))

(-79.99876436339626, -0.0121953057190993*cos(1.99876436339626*pi))

(63.99836575864295, 0.0161294574101027*cos(1.99836575864295*pi))

(-55.998253047962045, -0.0172418986339648*cos(1.99825304796205*pi))

(-39.99758749848229, -0.0238108915193316*cos(1.99758749848229*pi))

(-13.993665754266546, -0.0625247529468493*cos(1.99366575426655*pi))

(-45.997889080118895, -0.0208342495714923*cos(1.9978890801189*pi))

(-73.99866681093987, -0.0131581255561716*cos(1.99866681093987*pi))

(71.99855253456555, 0.0142860096929318*cos(1.99855253456555*pi))

(-59.99836575864295, -0.0161294574101027*cos(1.99836575864295*pi))

(53.99805146702257, 0.0192314898690811*cos(1.99805146702257*pi))

(-71.99863077792584, -0.0135137635586942*cos(1.99863077792584*pi))

(17.993665754266548, 0.0625247529468493*cos(1.99366575426655*pi))

(-65.99850996082668, -0.014706204600308*cos(1.99850996082668*pi))

(-23.996102641928523, -0.0384673046484715*cos(1.99610264192852*pi))

(-31.997019791037587, -0.0294143429673098*cos(1.99701979103759*pi))

(-25.9963810744755, -0.0357189022873998*cos(1.9963810744755*pi))

(-7.989861028618183, -0.100101492616892*cos(1.98986102861818*pi))

(7.983081334472861, 0.167137958536227*cos(1.98308133447286*pi))

(43.99758749848229, 0.0238108915193316*cos(1.99758749848229*pi))

(-37.99746686346282, -0.0250015833106043*cos(1.99746686346282*pi))

(-69.99859274300154, -0.0138891603558071*cos(1.99859274300154*pi))

(11.989861028618183, 0.100101492616892*cos(1.98986102861818*pi))

(-35.99733352832425, -0.026317636190302*cos(1.99733352832425*pi))

(-27.996622373693775, -0.0333370866740308*cos(1.99662237369377*pi))

(77.99866681093987, 0.0131581255561716*cos(1.99866681093987*pi))

(-1.9745618773827747, -0.251600058283278*cos(1.97456187738277*pi))

(-19.99539384872894, -0.0454640642889778*cos(1.99539384872894*pi))

(67.99846480674465, 0.0151518675915899*cos(1.99846480674465*pi))

(-5.9873214572994575, -0.125198416683996*cos(1.98732145729946*pi))

(-87.9988741996823, -0.0111112501005433*cos(1.99887419968231*pi))

(65.998416830397, 0.0156253865255779*cos(1.99841683039701*pi))

(-63.99846480674464, -0.0151518675915899*cos(1.99846480674464*pi))

(-51.99812363831861, -0.0185191620119624*cos(1.99812363831861*pi))

(85.99879378388556, 0.0119049328562125*cos(1.99879378388556*pi))

(45.997697165477256, 0.0227284622701719*cos(1.99769716547726*pi))

(49.997889080118895, 0.0208342495714923*cos(1.9978890801189*pi))

(-95.99896610309928, -0.0102041892865273*cos(1.99896610309928*pi))

(-47.99797352157236, -0.0200008106242253*cos(1.99797352157236*pi))

(25.995777788821496, 0.0416739981842078*cos(1.9957777888215*pi))

(99.99896610309928, 0.0102041892865273*cos(1.99896610309928*pi))

(-29.996833504122364, -0.0312530925871513*cos(1.99683350412236*pi))

(27.996102641928523, 0.0384673046484715*cos(1.99610264192852*pi))

(83.99876436339626, 0.0121953057190993*cos(1.99876436339626*pi))

(61.99831128191967, 0.0166671357682253*cos(1.99831128191967*pi))

(51.99797352157236, 0.0200008106242253*cos(1.99797352157236*pi))

(-57.99831128191967, -0.0166671357682253*cos(1.99831128191967*pi))

(3.9484545394207995, 0.513227267954253*cos(1.9484545394208*pi))

(55.99812363831861, 0.0185191620119624*cos(1.99812363831861*pi))

(73.99859274300154, 0.0138891603558071*cos(1.99859274300154*pi))

(9.987321457299458, 0.125198416683996*cos(1.98732145729946*pi))

(-75.99870099604381, -0.0128207263355671*cos(1.99870099604381*pi))

(13.991552603761589, 0.0833920371317317*cos(1.99155260376159*pi))

(47.997797295255594, 0.0217401714604091*cos(1.99779729525559*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:
$$x_{1} = -53.9981906541763$$
$$x_{2} = -11.9927602767546$$
$$x_{3} = -81.9987937838856$$
$$x_{4} = -85.99884861287$$
$$x_{5} = -17.9949330850375$$
$$x_{6} = -9.99155260376159$$
$$x_{7} = -49.9980514670226$$
$$x_{8} = 0.0515454605792004$$
$$x_{9} = -43.9977972952556$$
$$x_{10} = -91.9989221068275$$
$$x_{11} = -77.998733471837$$
$$x_{12} = -83.9988218359416$$
$$x_{13} = -21.9957777888215$$
$$x_{14} = -99.999006648613$$
$$x_{15} = -15.9943698713525$$
$$x_{16} = -41.9976971654773$$
$$x_{17} = -67.9985525345655$$
$$x_{18} = -89.9988986739972$$
$$x_{19} = -33.9971853759713$$
$$x_{20} = -93.998944563268$$
$$x_{21} = -97.9989867813195$$
$$x_{22} = -3.98308133447286$$
$$x_{23} = -61.998416830397$$
$$x_{24} = -79.9987643633963$$
$$x_{25} = -55.998253047962$$
$$x_{26} = -39.9975874984823$$
$$x_{27} = -13.9936657542665$$
$$x_{28} = -45.9978890801189$$
$$x_{29} = -73.9986668109399$$
$$x_{30} = -59.9983657586429$$
$$x_{31} = -71.9986307779258$$
$$x_{32} = -65.9985099608267$$
$$x_{33} = -23.9961026419285$$
$$x_{34} = -31.9970197910376$$
$$x_{35} = -25.9963810744755$$
$$x_{36} = -7.98986102861818$$
$$x_{37} = -37.9974668634628$$
$$x_{38} = -69.9985927430015$$
$$x_{39} = -35.9973335283242$$
$$x_{40} = -27.9966223736938$$
$$x_{41} = -1.97456187738277$$
$$x_{42} = -19.9953938487289$$
$$x_{43} = -5.98732145729946$$
$$x_{44} = -87.9988741996823$$
$$x_{45} = -63.9984648067446$$
$$x_{46} = -51.9981236383186$$
$$x_{47} = -95.9989661030993$$
$$x_{48} = -47.9979735215724$$
$$x_{49} = -29.9968335041224$$
$$x_{50} = -57.9983112819197$$
$$x_{51} = -75.9987009960438$$
Puntos máximos de la función:
$$x_{51} = 87.9988218359416$$
$$x_{51} = 31.9966223736938$$
$$x_{51} = 15.9927602767546$$
$$x_{51} = 35.9970197910376$$
$$x_{51} = 19.9943698713525$$
$$x_{51} = 29.9963810744755$$
$$x_{51} = 81.998733471837$$
$$x_{51} = 79.9987009960438$$
$$x_{51} = 59.998253047962$$
$$x_{51} = 89.99884861287$$
$$x_{51} = 69.9985099608267$$
$$x_{51} = 33.9968335041224$$
$$x_{51} = 97.998944563268$$
$$x_{51} = 93.9988986739972$$
$$x_{51} = 41.9974668634628$$
$$x_{51} = 23.9953938487289$$
$$x_{51} = 39.9973335283242$$
$$x_{51} = 5.97456187738277$$
$$x_{51} = 75.9986307779258$$
$$x_{51} = 57.9981906541763$$
$$x_{51} = 37.9971853759713$$
$$x_{51} = 21.9949330850375$$
$$x_{51} = 95.9989221068275$$
$$x_{51} = 91.9988741996823$$
$$x_{51} = 63.9983657586429$$
$$x_{51} = 71.9985525345655$$
$$x_{51} = 53.9980514670226$$
$$x_{51} = 17.9936657542665$$
$$x_{51} = 7.98308133447286$$
$$x_{51} = 43.9975874984823$$
$$x_{51} = 11.9898610286182$$
$$x_{51} = 77.9986668109399$$
$$x_{51} = 67.9984648067446$$
$$x_{51} = 65.998416830397$$
$$x_{51} = 85.9987937838856$$
$$x_{51} = 45.9976971654773$$
$$x_{51} = 49.9978890801189$$
$$x_{51} = 25.9957777888215$$
$$x_{51} = 99.9989661030993$$
$$x_{51} = 27.9961026419285$$
$$x_{51} = 83.9987643633963$$
$$x_{51} = 61.9983112819197$$
$$x_{51} = 51.9979735215724$$
$$x_{51} = 3.9484545394208$$
$$x_{51} = 55.9981236383186$$
$$x_{51} = 73.9985927430015$$
$$x_{51} = 9.98732145729946$$
$$x_{51} = 13.9915526037616$$
$$x_{51} = 47.9977972952556$$
Decrece en los intervalos
$$\left[0.0515454605792004, 3.9484545394208\right]$$
Crece en los intervalos
$$\left(-\infty, -99.999006648613\right]$$