Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • -x^2-2x+1
  • (x+1)(x-2)^2
  • 6x-2x^2
  • 9^(1/(x-3))

  • Identical expressions

  • cos(pi*x)/(x- two)
  • co sinus of e of ( Pi multiply by x) divide by (x minus 2)
  • co sinus of e of ( Pi multiply by x) divide by (x minus two)
  • cos(pix)/(x-2)
  • cospix/x-2
  • cos(pi*x) divide by (x-2)

  • Similar expressions

  • cos(pi*x)/(x+2)
Plotting a function graph/ cos(pi*x)/(x-2)

Graphing y = cos(pi*x)/(x-2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(pi*x)
f(x) = ---------
         x - 2
f(x)=cos⁡(πx)x−2f{\left(x \right)} = \frac{\cos{\left(\pi x \right)}}{x - 2} 
f = cos(pi*x)/(x - 2)
The graph of the function
02468-8-6-4-2-1010-5050
The domain of the function
The points at which the function is not precisely defined:
x1=2x_{1} = 2
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
cos⁡(πx)x−2=0\frac{\cos{\left(\pi x \right)}}{x - 2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=12x_{1} = \frac{1}{2}
x2=32x_{2} = \frac{3}{2}
Numerical solution
x1=54.5x_{1} = 54.5
x2=12.5x_{2} = 12.5
x3=−43.5x_{3} = -43.5
x4=−13.5x_{4} = -13.5
x5=82.5x_{5} = 82.5
x6=−75.5x_{6} = -75.5
x7=62.5x_{7} = 62.5
x8=46.5x_{8} = 46.5
x9=32.5x_{9} = 32.5
x10=42.5x_{10} = 42.5
x11=26.5x_{11} = 26.5
x12=4.5x_{12} = 4.5
x13=48.5x_{13} = 48.5
x14=−7.5x_{14} = -7.5
x15=−41.5x_{15} = -41.5
x16=72.5x_{16} = 72.5
x17=96.5x_{17} = 96.5
x18=−55.5x_{18} = -55.5
x19=56.5x_{19} = 56.5
x20=−37.5x_{20} = -37.5
x21=84.5x_{21} = 84.5
x22=28.5x_{22} = 28.5
x23=70.5x_{23} = 70.5
x24=−3.5x_{24} = -3.5
x25=−51.5x_{25} = -51.5
x26=10.5x_{26} = 10.5
x27=8.5x_{27} = 8.5
x28=−89.5x_{28} = -89.5
x29=74.5x_{29} = 74.5
x30=−81.5x_{30} = -81.5
x31=−71.5x_{31} = -71.5
x32=−87.5x_{32} = -87.5
x33=−97.5x_{33} = -97.5
x34=68.5x_{34} = 68.5
x35=−53.5x_{35} = -53.5
x36=34.5x_{36} = 34.5
x37=94.5x_{37} = 94.5
x38=−93.5x_{38} = -93.5
x39=−31.5x_{39} = -31.5
x40=60.5x_{40} = 60.5
x41=−91.5x_{41} = -91.5
x42=−65.5x_{42} = -65.5
x43=−73.5x_{43} = -73.5
x44=−5.5x_{44} = -5.5
x45=−23.5x_{45} = -23.5
x46=−95.5x_{46} = -95.5
x47=−67.5x_{47} = -67.5
x48=78.5x_{48} = 78.5
x49=−15.5x_{49} = -15.5
x50=−19.5x_{50} = -19.5
x51=14.5x_{51} = 14.5
x52=22.5x_{52} = 22.5
x53=−85.5x_{53} = -85.5
x54=0.5x_{54} = 0.5
x55=−99.5x_{55} = -99.5
x56=92.5x_{56} = 92.5
x57=−21.5x_{57} = -21.5
x58=−35.5x_{58} = -35.5
x59=90.5x_{59} = 90.5
x60=−63.5x_{60} = -63.5
x61=66.5x_{61} = 66.5
x62=−27.5x_{62} = -27.5
x63=18.5x_{63} = 18.5
x64=44.5x_{64} = 44.5
x65=−45.5x_{65} = -45.5
x66=−79.5x_{66} = -79.5
x67=−9.5x_{67} = -9.5
x68=−59.5x_{68} = -59.5
x69=16.5x_{69} = 16.5
x70=−33.5x_{70} = -33.5
x71=52.5x_{71} = 52.5
x72=−61.5x_{72} = -61.5
x73=6.5x_{73} = 6.5
x74=30.5x_{74} = 30.5
x75=58.5x_{75} = 58.5
x76=64.5x_{76} = 64.5
x77=−49.5x_{77} = -49.5
x78=−77.5x_{78} = -77.5
x79=86.5x_{79} = 86.5
x80=80.5x_{80} = 80.5
x81=98.5x_{81} = 98.5
x82=38.5x_{82} = 38.5
x83=100.5x_{83} = 100.5
x84=88.5x_{84} = 88.5
x85=−57.5x_{85} = -57.5
x86=−1.5x_{86} = -1.5
x87=36.5x_{87} = 36.5
x88=−69.5x_{88} = -69.5
x89=40.5x_{89} = 40.5
x90=76.5x_{90} = 76.5
x91=20.5x_{91} = 20.5
x92=50.5x_{92} = 50.5
x93=−11.5x_{93} = -11.5
x94=−29.5x_{94} = -29.5
x95=−39.5x_{95} = -39.5
x96=−47.5x_{96} = -47.5
x97=−17.5x_{97} = -17.5
x98=−25.5x_{98} = -25.5
x99=24.5x_{99} = 24.5
x100=−83.5x_{100} = -83.5
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(pi*x)/(x - 2).
cos⁡(0π)−2\frac{\cos{\left(0 \pi \right)}}{-2}
The result:
f(0)=−12f{\left(0 \right)} = - \frac{1}{2}
The point:
(0, -1/2)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−πsin⁡(πx)x−2−cos⁡(πx)(x−2)2=0- \frac{\pi \sin{\left(\pi x \right)}}{x - 2} - \frac{\cos{\left(\pi x \right)}}{\left(x - 2\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=−33.9971853759713x_{1} = -33.9971853759713
x2=−65.9985099608267x_{2} = -65.9985099608267
x3=43.9975874984823x_{3} = 43.9975874984823
x4=57.9981906541763x_{4} = 57.9981906541763
x5=−57.9983112819197x_{5} = -57.9983112819197
x6=77.9986668109399x_{6} = 77.9986668109399
x7=37.9971853759713x_{7} = 37.9971853759713
x8=89.99884861287x_{8} = 89.99884861287
x9=17.9936657542665x_{9} = 17.9936657542665
x10=−13.9936657542665x_{10} = -13.9936657542665
x11=39.9973335283242x_{11} = 39.9973335283242
x12=79.9987009960438x_{12} = 79.9987009960438
x13=61.9983112819197x_{13} = 61.9983112819197
x14=−63.9984648067446x_{14} = -63.9984648067446
x15=67.9984648067446x_{15} = 67.9984648067446
x16=−23.9961026419285x_{16} = -23.9961026419285
x17=−49.9980514670226x_{17} = -49.9980514670226
x18=−11.9927602767546x_{18} = -11.9927602767546
x19=−91.9989221068275x_{19} = -91.9989221068275
x20=75.9986307779258x_{20} = 75.9986307779258
x21=15.9927602767546x_{21} = 15.9927602767546
x22=−77.998733471837x_{22} = -77.998733471837
x23=3.9484545394208x_{23} = 3.9484545394208
x24=−67.9985525345655x_{24} = -67.9985525345655
x25=−17.9949330850375x_{25} = -17.9949330850375
x26=91.9988741996823x_{26} = 91.9988741996823
x27=−39.9975874984823x_{27} = -39.9975874984823
x28=−99.999006648613x_{28} = -99.999006648613
x29=−71.9986307779258x_{29} = -71.9986307779258
x30=71.9985525345655x_{30} = 71.9985525345655
x31=−87.9988741996823x_{31} = -87.9988741996823
x32=−61.998416830397x_{32} = -61.998416830397
x33=−95.9989661030993x_{33} = -95.9989661030993
x34=29.9963810744755x_{34} = 29.9963810744755
x35=21.9949330850375x_{35} = 21.9949330850375
x36=7.98308133447286x_{36} = 7.98308133447286
x37=−93.998944563268x_{37} = -93.998944563268
x38=87.9988218359416x_{38} = 87.9988218359416
x39=35.9970197910376x_{39} = 35.9970197910376
x40=−37.9974668634628x_{40} = -37.9974668634628
x41=−83.9988218359416x_{41} = -83.9988218359416
x42=−97.9989867813195x_{42} = -97.9989867813195
x43=−21.9957777888215x_{43} = -21.9957777888215
x44=−73.9986668109399x_{44} = -73.9986668109399
x45=−1.97456187738277x_{45} = -1.97456187738277
x46=−47.9979735215724x_{46} = -47.9979735215724
x47=−27.9966223736938x_{47} = -27.9966223736938
x48=73.9985927430015x_{48} = 73.9985927430015
x49=5.97456187738277x_{49} = 5.97456187738277
x50=53.9980514670226x_{50} = 53.9980514670226
x51=11.9898610286182x_{51} = 11.9898610286182
x52=−35.9973335283242x_{52} = -35.9973335283242
x53=−69.9985927430015x_{53} = -69.9985927430015
x54=−3.98308133447286x_{54} = -3.98308133447286
x55=27.9961026419285x_{55} = 27.9961026419285
x56=47.9977972952556x_{56} = 47.9977972952556
x57=−29.9968335041224x_{57} = -29.9968335041224
x58=81.998733471837x_{58} = 81.998733471837
x59=31.9966223736938x_{59} = 31.9966223736938
x60=−81.9987937838856x_{60} = -81.9987937838856
x61=−9.99155260376159x_{61} = -9.99155260376159
x62=−5.98732145729946x_{62} = -5.98732145729946
x63=−85.99884861287x_{63} = -85.99884861287
x64=−55.998253047962x_{64} = -55.998253047962
x65=−53.9981906541763x_{65} = -53.9981906541763
x66=9.98732145729946x_{66} = 9.98732145729946
x67=25.9957777888215x_{67} = 25.9957777888215
x68=49.9978890801189x_{68} = 49.9978890801189
x69=−51.9981236383186x_{69} = -51.9981236383186
x70=−45.9978890801189x_{70} = -45.9978890801189
x71=41.9974668634628x_{71} = 41.9974668634628
x72=93.9988986739972x_{72} = 93.9988986739972
x73=0.0515454605792004x_{73} = 0.0515454605792004
x74=−31.9970197910376x_{74} = -31.9970197910376
x75=45.9976971654773x_{75} = 45.9976971654773
x76=97.998944563268x_{76} = 97.998944563268
x77=69.9985099608267x_{77} = 69.9985099608267
x78=95.9989221068275x_{78} = 95.9989221068275
x79=63.9983657586429x_{79} = 63.9983657586429
x80=−15.9943698713525x_{80} = -15.9943698713525
x81=−89.9988986739972x_{81} = -89.9988986739972
x82=23.9953938487289x_{82} = 23.9953938487289
x83=83.9987643633963x_{83} = 83.9987643633963
x84=19.9943698713525x_{84} = 19.9943698713525
x85=85.9987937838856x_{85} = 85.9987937838856
x86=65.998416830397x_{86} = 65.998416830397
x87=−75.9987009960438x_{87} = -75.9987009960438
x88=−19.9953938487289x_{88} = -19.9953938487289
x89=−79.9987643633963x_{89} = -79.9987643633963
x90=−43.9977972952556x_{90} = -43.9977972952556
x91=51.9979735215724x_{91} = 51.9979735215724
x92=13.9915526037616x_{92} = 13.9915526037616
x93=59.998253047962x_{93} = 59.998253047962
x94=−25.9963810744755x_{94} = -25.9963810744755
x95=−59.9983657586429x_{95} = -59.9983657586429
x96=55.9981236383186x_{96} = 55.9981236383186
x97=−7.98986102861818x_{97} = -7.98986102861818
x98=33.9968335041224x_{98} = 33.9968335041224
x99=99.9989661030993x_{99} = 99.9989661030993
x100=−41.9976971654773x_{100} = -41.9976971654773
The values of the extrema at the points:
(-33.99718537597127, -0.0277799497253893*cos(1.99718537597127*pi))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(47.997797295255594, 0.0217401714604091*cos(1.99779729525559*pi))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=−33.9971853759713x_{1} = -33.9971853759713
x2=−65.9985099608267x_{2} = -65.9985099608267
x3=−57.9983112819197x_{3} = -57.9983112819197
x4=−13.9936657542665x_{4} = -13.9936657542665
x5=−63.9984648067446x_{5} = -63.9984648067446
x6=−23.9961026419285x_{6} = -23.9961026419285
x7=−49.9980514670226x_{7} = -49.9980514670226
x8=−11.9927602767546x_{8} = -11.9927602767546
x9=−91.9989221068275x_{9} = -91.9989221068275
x10=−77.998733471837x_{10} = -77.998733471837
x11=−67.9985525345655x_{11} = -67.9985525345655
x12=−17.9949330850375x_{12} = -17.9949330850375
x13=−39.9975874984823x_{13} = -39.9975874984823
x14=−99.999006648613x_{14} = -99.999006648613
x15=−71.9986307779258x_{15} = -71.9986307779258
x16=−87.9988741996823x_{16} = -87.9988741996823
x17=−61.998416830397x_{17} = -61.998416830397
x18=−95.9989661030993x_{18} = -95.9989661030993
x19=−93.998944563268x_{19} = -93.998944563268
x20=−37.9974668634628x_{20} = -37.9974668634628
x21=−83.9988218359416x_{21} = -83.9988218359416
x22=−97.9989867813195x_{22} = -97.9989867813195
x23=−21.9957777888215x_{23} = -21.9957777888215
x24=−73.9986668109399x_{24} = -73.9986668109399
x25=−1.97456187738277x_{25} = -1.97456187738277
x26=−47.9979735215724x_{26} = -47.9979735215724
x27=−27.9966223736938x_{27} = -27.9966223736938
x28=−35.9973335283242x_{28} = -35.9973335283242
x29=−69.9985927430015x_{29} = -69.9985927430015
x30=−3.98308133447286x_{30} = -3.98308133447286
x31=−29.9968335041224x_{31} = -29.9968335041224
x32=−81.9987937838856x_{32} = -81.9987937838856
x33=−9.99155260376159x_{33} = -9.99155260376159
x34=−5.98732145729946x_{34} = -5.98732145729946
x35=−85.99884861287x_{35} = -85.99884861287
x36=−55.998253047962x_{36} = -55.998253047962
x37=−53.9981906541763x_{37} = -53.9981906541763
x38=−51.9981236383186x_{38} = -51.9981236383186
x39=−45.9978890801189x_{39} = -45.9978890801189
x40=0.0515454605792004x_{40} = 0.0515454605792004
x41=−31.9970197910376x_{41} = -31.9970197910376
x42=−15.9943698713525x_{42} = -15.9943698713525
x43=−89.9988986739972x_{43} = -89.9988986739972
x44=−75.9987009960438x_{44} = -75.9987009960438
x45=−19.9953938487289x_{45} = -19.9953938487289
x46=−79.9987643633963x_{46} = -79.9987643633963
x47=−43.9977972952556x_{47} = -43.9977972952556
x48=−25.9963810744755x_{48} = -25.9963810744755
x49=−59.9983657586429x_{49} = -59.9983657586429
x50=−7.98986102861818x_{50} = -7.98986102861818
x51=−41.9976971654773x_{51} = -41.9976971654773
Maxima of the function at points:
x51=43.9975874984823x_{51} = 43.9975874984823
x51=57.9981906541763x_{51} = 57.9981906541763
x51=77.9986668109399x_{51} = 77.9986668109399
x51=37.9971853759713x_{51} = 37.9971853759713
x51=89.99884861287x_{51} = 89.99884861287
x51=17.9936657542665x_{51} = 17.9936657542665
x51=39.9973335283242x_{51} = 39.9973335283242
x51=79.9987009960438x_{51} = 79.9987009960438
x51=61.9983112819197x_{51} = 61.9983112819197
x51=67.9984648067446x_{51} = 67.9984648067446
x51=75.9986307779258x_{51} = 75.9986307779258
x51=15.9927602767546x_{51} = 15.9927602767546
x51=3.9484545394208x_{51} = 3.9484545394208
x51=91.9988741996823x_{51} = 91.9988741996823
x51=71.9985525345655x_{51} = 71.9985525345655
x51=29.9963810744755x_{51} = 29.9963810744755
x51=21.9949330850375x_{51} = 21.9949330850375
x51=7.98308133447286x_{51} = 7.98308133447286
x51=87.9988218359416x_{51} = 87.9988218359416
x51=35.9970197910376x_{51} = 35.9970197910376
x51=73.9985927430015x_{51} = 73.9985927430015
x51=5.97456187738277x_{51} = 5.97456187738277
x51=53.9980514670226x_{51} = 53.9980514670226
x51=11.9898610286182x_{51} = 11.9898610286182
x51=27.9961026419285x_{51} = 27.9961026419285
x51=47.9977972952556x_{51} = 47.9977972952556
x51=81.998733471837x_{51} = 81.998733471837
x51=31.9966223736938x_{51} = 31.9966223736938
x51=9.98732145729946x_{51} = 9.98732145729946
x51=25.9957777888215x_{51} = 25.9957777888215
x51=49.9978890801189x_{51} = 49.9978890801189
x51=41.9974668634628x_{51} = 41.9974668634628
x51=93.9988986739972x_{51} = 93.9988986739972
x51=45.9976971654773x_{51} = 45.9976971654773
x51=97.998944563268x_{51} = 97.998944563268
x51=69.9985099608267x_{51} = 69.9985099608267
x51=95.9989221068275x_{51} = 95.9989221068275
x51=63.9983657586429x_{51} = 63.9983657586429
x51=23.9953938487289x_{51} = 23.9953938487289
x51=83.9987643633963x_{51} = 83.9987643633963
x51=19.9943698713525x_{51} = 19.9943698713525
x51=85.9987937838856x_{51} = 85.9987937838856
x51=65.998416830397x_{51} = 65.998416830397
x51=51.9979735215724x_{51} = 51.9979735215724
x51=13.9915526037616x_{51} = 13.9915526037616
x51=59.998253047962x_{51} = 59.998253047962
x51=55.9981236383186x_{51} = 55.9981236383186
x51=33.9968335041224x_{51} = 33.9968335041224
x51=99.9989661030993x_{51} = 99.9989661030993
Decreasing at intervals
[0.0515454605792004,3.9484545394208]\left[0.0515454605792004, 3.9484545394208\right]
Increasing at intervals
(−∞,−99.999006648613]\left(-\infty, -99.999006648613\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−π2cos⁡(πx)+2πsin⁡(πx)x−2+2cos⁡(πx)(x−2)2x−2=0\frac{- \pi^{2} \cos{\left(\pi x \right)} + \frac{2 \pi \sin{\left(\pi x \right)}}{x - 2} + \frac{2 \cos{\left(\pi x \right)}}{\left(x - 2\right)^{2}}}{x - 2} = 0
Solve this equation
The roots of this equation
x1=−49.4960647958338x_{1} = -49.4960647958338
x2=−23.4920499360659x_{2} = -23.4920499360659
x3=82.4974825985472x_{3} = 82.4974825985472
x4=18.487706434622x_{4} = 18.487706434622
x5=−95.4979215576703x_{5} = -95.4979215576703
x6=−87.4977357630958x_{6} = -87.4977357630958
x7=56.4962814531436x_{7} = 56.4962814531436
x8=66.496858053806x_{8} = 66.496858053806
x9=100.497942659789x_{9} = 100.497942659789
x10=68.496952560156x_{10} = 68.496952560156
x11=−55.4964754968732x_{11} = -55.4964754968732
x12=94.4978092025222x_{12} = 94.4978092025222
x13=−93.4978780275295x_{13} = -93.4978780275295
x14=−63.4969060285399x_{14} = -63.4969060285399
x15=6.4543529700251x_{15} = 6.4543529700251
x16=54.4961397669456x_{16} = 54.4961397669456
x17=−97.4979633377095x_{17} = -97.4979633377095
x18=52.4959868546431x_{18} = 52.4959868546431
x19=−75.4973851421441x_{19} = -75.4973851421441
x20=−91.4978326349676x_{20} = -91.4978326349676
x21=48.495641554955x_{21} = 48.495641554955
x22=−43.4955457402116x_{22} = -43.4955457402116
x23=−53.4963484658526x_{23} = -53.4963484658526
x24=−31.4939495171458x_{24} = -31.4939495171458
x25=−77.4974509304293x_{25} = -77.4974509304293
x26=64.4967574978323x_{26} = 64.4967574978323
x27=96.4978555714426x_{27} = 96.4978555714426
x28=−37.4948689248164x_{28} = -37.4948689248164
x29=38.4944470279643x_{29} = 38.4944470279643
x30=−35.494595164929x_{30} = -35.494595164929
x31=−57.4965939859678x_{31} = -57.4965939859678
x32=−33.4942905400287x_{32} = -33.4942905400287
x33=−21.4913726980341x_{33} = -21.4913726980341
x34=22.490108643939x_{34} = 22.490108643939
x35=−47.4959057632916x_{35} = -47.4959057632916
x36=40.4947356021421x_{36} = 40.4947356021421
x37=−89.4977852578432x_{37} = -89.4977852578432
x38=88.4976572290714x_{38} = 88.4976572290714
x39=−9.48234279411391x_{39} = -9.48234279411391
x40=−85.4976840054972x_{40} = -85.4976840054972
x41=10.476069974762x_{41} = 10.476069974762
x42=42.4949956603161x_{42} = 42.4949956603161
x43=−81.4975730512497x_{43} = -81.4975730512497
x44=12.4806532247953x_{44} = 12.4806532247953
x45=8.46862260293437x_{45} = 8.46862260293437
x46=58.4964131058168x_{46} = 58.4964131058168
x47=−15.4884102041403x_{47} = -15.4884102041403
x48=60.4965357543878x_{48} = 60.4965357543878
x49=32.4933540572253x_{49} = 32.4933540572253
x50=−29.4935651564022x_{50} = -29.4935651564022
x51=98.4979000181643x_{51} = 98.4979000181643
x52=−79.4975134894374x_{52} = -79.4975134894374
x53=−39.4951162851327x_{53} = -39.4951162851327
x54=−25.4926285522864x_{54} = -25.4926285522864
x55=−73.4973158679038x_{55} = -73.4973158679038
x56=80.4974184553755x_{56} = 80.4974184553755
x57=90.4977101767426x_{57} = 90.4977101767426
x58=78.4973509578595x_{58} = 78.4973509578595
x59=70.4970415469044x_{59} = 70.4970415469044
x60=−71.4972428230858x_{60} = -71.4972428230858
x61=−19.4905692587914x_{61} = -19.4905692587914
x62=−65.4969977128086x_{62} = -65.4969977128086
x63=30.4928873749286x_{63} = 30.4928873749286
x64=74.4972047891757x_{64} = 74.4972047891757
x65=26.4917251569433x_{65} = 26.4917251569433
x66=76.4972798358001x_{66} = 76.4972798358001
x67=−83.4976298262279x_{67} = -83.4976298262279
x68=34.4937632545603x_{68} = 34.4937632545603
x69=−99.4980034711124x_{69} = -99.4980034711124
x70=86.4976017747235x_{70} = 86.4976017747235
x71=−67.4970841193725x_{71} = -67.4970841193725
x72=−7.47860502542155x_{72} = -7.47860502542155
x73=4.41514421970787x_{73} = 4.41514421970787
x74=72.4971254839111x_{74} = 72.4971254839111
x75=62.4966502921034x_{75} = 62.4966502921034
x76=24.4909888612868x_{76} = 24.4909888612868
x77=28.4923501740906x_{77} = 28.4923501740906
x78=−17.4896006904042x_{78} = -17.4896006904042
x79=14.4837604918852x_{79} = 14.4837604918852
x80=84.4975436313672x_{80} = 84.4975436313672
x81=36.4941249743177x_{81} = 36.4941249743177
x82=16.4860066639937x_{82} = 16.4860066639937
x83=20.4890376991024x_{83} = 20.4890376991024
x84=−13.4869115667323x_{84} = -13.4869115667323
x85=50.4958213268372x_{85} = 50.4958213268372
x86=−41.4953408894394x_{86} = -41.4953408894394
x87=−11.4849671424621x_{87} = -11.4849671424621
x88=−2.4543529700251x_{88} = -2.4543529700251
x89=−69.4971656912136x_{89} = -69.4971656912136
x90=44.4952312308764x_{90} = 44.4952312308764
x91=−5.47285012518195x_{91} = -5.47285012518195
x92=−61.4968085677577x_{92} = -61.4968085677577
x93=−27.4931286333261x_{93} = -27.4931286333261
x94=−45.495733333912x_{94} = -45.495733333912
x95=46.4954456174624x_{95} = 46.4954456174624
x96=92.4977607839483x_{96} = 92.4977607839483
x97=−59.496704766746x_{97} = -59.496704766746
x98=−51.4962119345899x_{98} = -51.4962119345899
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=2x_{1} = 2

lim⁡x→2−(−π2cos⁡(πx)+2πsin⁡(πx)x−2+2cos⁡(πx)(x−2)2x−2)=−∞\lim_{x \to 2^-}\left(\frac{- \pi^{2} \cos{\left(\pi x \right)} + \frac{2 \pi \sin{\left(\pi x \right)}}{x - 2} + \frac{2 \cos{\left(\pi x \right)}}{\left(x - 2\right)^{2}}}{x - 2}\right) = -\infty
lim⁡x→2+(−π2cos⁡(πx)+2πsin⁡(πx)x−2+2cos⁡(πx)(x−2)2x−2)=∞\lim_{x \to 2^+}\left(\frac{- \pi^{2} \cos{\left(\pi x \right)} + \frac{2 \pi \sin{\left(\pi x \right)}}{x - 2} + \frac{2 \cos{\left(\pi x \right)}}{\left(x - 2\right)^{2}}}{x - 2}\right) = \infty
- the limits are not equal, so
x1=2x_{1} = 2
- is an inflection point

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[100.497942659789,∞)\left[100.497942659789, \infty\right)
Convex at the intervals
[−5.47285012518195,−2.4543529700251]\left[-5.47285012518195, -2.4543529700251\right]
Vertical asymptotes
Have:
x1=2x_{1} = 2
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(cos⁡(πx)x−2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\pi x \right)}}{x - 2}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(cos⁡(πx)x−2)=0\lim_{x \to \infty}\left(\frac{\cos{\left(\pi x \right)}}{x - 2}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(pi*x)/(x - 2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(πx)x(x−2))=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\pi x \right)}}{x \left(x - 2\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(πx)x(x−2))=0\lim_{x \to \infty}\left(\frac{\cos{\left(\pi x \right)}}{x \left(x - 2\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
cos⁡(πx)x−2=cos⁡(πx)−x−2\frac{\cos{\left(\pi x \right)}}{x - 2} = \frac{\cos{\left(\pi x \right)}}{- x - 2}
- No
cos⁡(πx)x−2=−cos⁡(πx)−x−2\frac{\cos{\left(\pi x \right)}}{x - 2} = - \frac{\cos{\left(\pi x \right)}}{- x - 2}
- No
so, the function
not is
neither even, nor odd
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