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Plotting a function graph/ x*cot(x/2)

Graphing y = x*cot(x/2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            /x\
f(x) = x*cot|-|
            \2/
f(x)=xcot⁡(x2)f{\left(x \right)} = x \cot{\left(\frac{x}{2} \right)} 
f = x*cot(x/2)
The graph of the function
02468-8-6-4-2-1010-1000010000
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:
xcot⁡(x2)=0x \cot{\left(\frac{x}{2} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=πx_{1} = \pi
Numerical solution
x1=72.2566310325652x_{1} = 72.2566310325652
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=3.14159265358979x_{3} = 3.14159265358979
x4=28.2743338823081x_{4} = 28.2743338823081
x5=65.9734457253857x_{5} = 65.9734457253857
x6=−9.42477796076938x_{6} = -9.42477796076938
x7=40.8407044966673x_{7} = 40.8407044966673
x8=−3.14159265358979x_{8} = -3.14159265358979
x9=−15.707963267949x_{9} = -15.707963267949
x10=59.6902604182061x_{10} = 59.6902604182061
x11=9.42477796076938x_{11} = 9.42477796076938
x12=−53.4070751110265x_{12} = -53.4070751110265
x13=−47.1238898038469x_{13} = -47.1238898038469
x14=−84.8230016469244x_{14} = -84.8230016469244
x15=21.9911485751286x_{15} = 21.9911485751286
x16=−72.2566310325652x_{16} = -72.2566310325652
x17=34.5575191894877x_{17} = 34.5575191894877
x18=−21.9911485751286x_{18} = -21.9911485751286
x19=−65.9734457253857x_{19} = -65.9734457253857
x20=53.4070751110265x_{20} = 53.4070751110265
x21=15.707963267949x_{21} = 15.707963267949
x22=−28.2743338823081x_{22} = -28.2743338823081
x23=−91.106186954104x_{23} = -91.106186954104
x24=47.1238898038469x_{24} = 47.1238898038469
x25=97.3893722612836x_{25} = 97.3893722612836
x26=78.5398163397448x_{26} = 78.5398163397448
x27=−78.5398163397448x_{27} = -78.5398163397448
x28=−40.8407044966673x_{28} = -40.8407044966673
x29=−97.3893722612836x_{29} = -97.3893722612836
x30=84.8230016469244x_{30} = 84.8230016469244
x31=91.106186954104x_{31} = 91.106186954104
x32=−34.5575191894877x_{32} = -34.5575191894877
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*cot(x/2).
0cot⁡(02)0 \cot{\left(\frac{0}{2} \right)}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
x(−cot⁡2(x2)2−12)+cot⁡(x2)=0x \left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}\right) + \cot{\left(\frac{x}{2} \right)} = 0
Solve this equation
The roots of this equation
x1=1.52707904860319⋅10−17x_{1} = 1.52707904860319 \cdot 10^{-17}
x2=2.23576865397504⋅10−17x_{2} = 2.23576865397504 \cdot 10^{-17}
x3=1.59668259179644⋅10−16x_{3} = 1.59668259179644 \cdot 10^{-16}
x4=−1.15725582418354⋅10−16x_{4} = -1.15725582418354 \cdot 10^{-16}
x5=−1.03125774822142⋅10−16x_{5} = -1.03125774822142 \cdot 10^{-16}
x6=3.26384634187708⋅10−16x_{6} = 3.26384634187708 \cdot 10^{-16}
x7=3.8991968647723⋅10−15x_{7} = 3.8991968647723 \cdot 10^{-15}
x8=−1.14081209393754⋅10−18x_{8} = -1.14081209393754 \cdot 10^{-18}
x9=−9.7885699769805⋅10−17x_{9} = -9.7885699769805 \cdot 10^{-17}
x10=−5.93046655993782⋅10−17x_{10} = -5.93046655993782 \cdot 10^{-17}
x11=1.72285514463146⋅10−14x_{11} = 1.72285514463146 \cdot 10^{-14}
x12=2.73073920295658⋅10−18x_{12} = 2.73073920295658 \cdot 10^{-18}
x13=−1.27878285278132⋅10−14x_{13} = -1.27878285278132 \cdot 10^{-14}
x14=−8.17704180853909⋅10−16x_{14} = -8.17704180853909 \cdot 10^{-16}
x15=1.42341105080165⋅10−17x_{15} = 1.42341105080165 \cdot 10^{-17}
x16=1.11719724635132⋅10−17x_{16} = 1.11719724635132 \cdot 10^{-17}
x17=9.94846102116844⋅10−18x_{17} = 9.94846102116844 \cdot 10^{-18}
x18=9.63403898273477⋅10−17x_{18} = 9.63403898273477 \cdot 10^{-17}
x19=3.81785798127671⋅10−15x_{19} = 3.81785798127671 \cdot 10^{-15}
x20=−3.86068492929349⋅10−16x_{20} = -3.86068492929349 \cdot 10^{-16}
x21=1.25842485455245⋅10−16x_{21} = 1.25842485455245 \cdot 10^{-16}
x22=−2.3951484385737⋅10−16x_{22} = -2.3951484385737 \cdot 10^{-16}
x23=−4.79273379738631⋅10−19x_{23} = -4.79273379738631 \cdot 10^{-19}
x24=2.0736465771844⋅10−19x_{24} = 2.0736465771844 \cdot 10^{-19}
x25=−1.47369389671908⋅10−13x_{25} = -1.47369389671908 \cdot 10^{-13}
x26=−6.39496545398449⋅10−18x_{26} = -6.39496545398449 \cdot 10^{-18}
x27=−5.19078680712148⋅10−17x_{27} = -5.19078680712148 \cdot 10^{-17}
x28=5.4218915002628⋅10−17x_{28} = 5.4218915002628 \cdot 10^{-17}
x29=−3.84488604463226⋅10−17x_{29} = -3.84488604463226 \cdot 10^{-17}
x30=2.19371504995755⋅10−15x_{30} = 2.19371504995755 \cdot 10^{-15}
x31=1.36554056592777⋅10−15x_{31} = 1.36554056592777 \cdot 10^{-15}
x32=6.16366879916214⋅10−18x_{32} = 6.16366879916214 \cdot 10^{-18}
x33=−1.9680036684828⋅10−15x_{33} = -1.9680036684828 \cdot 10^{-15}
x34=−1.08756072736682⋅10−15x_{34} = -1.08756072736682 \cdot 10^{-15}
x35=−2.66945581377393⋅10−17x_{35} = -2.66945581377393 \cdot 10^{-17}
x36=−1.46479077536905⋅10−18x_{36} = -1.46479077536905 \cdot 10^{-18}
x37=1.67937364663135⋅10−15x_{37} = 1.67937364663135 \cdot 10^{-15}
x38=8.43698087053401⋅10−15x_{38} = 8.43698087053401 \cdot 10^{-15}
x39=2.83986751875031⋅10−15x_{39} = 2.83986751875031 \cdot 10^{-15}
x40=−5.05569600354816⋅10−17x_{40} = -5.05569600354816 \cdot 10^{-17}
x41=−1.9863529512231⋅10−19x_{41} = -1.9863529512231 \cdot 10^{-19}
x42=−2.64163878352735⋅10−16x_{42} = -2.64163878352735 \cdot 10^{-16}
x43=1.90677010489577⋅10−17x_{43} = 1.90677010489577 \cdot 10^{-17}
x44=−1.36222961689762⋅10−19x_{44} = -1.36222961689762 \cdot 10^{-19}
x45=−1.11417406398154⋅10−15x_{45} = -1.11417406398154 \cdot 10^{-15}
x46=4.50932681131416⋅10−14x_{46} = 4.50932681131416 \cdot 10^{-14}
x47=2.1641423377405⋅10−17x_{47} = 2.1641423377405 \cdot 10^{-17}
x48=−3.80119294448226⋅10−16x_{48} = -3.80119294448226 \cdot 10^{-16}
x49=6.55183592229414⋅10−19x_{49} = 6.55183592229414 \cdot 10^{-19}
x50=8.7035549441581⋅10−15x_{50} = 8.7035549441581 \cdot 10^{-15}
x51=1.04949196674696⋅10−16x_{51} = 1.04949196674696 \cdot 10^{-16}
x52=−1.37583055851796⋅10−14x_{52} = -1.37583055851796 \cdot 10^{-14}
x53=−1.1823647648489⋅10−15x_{53} = -1.1823647648489 \cdot 10^{-15}
x54=−3.87889712429163⋅10−15x_{54} = -3.87889712429163 \cdot 10^{-15}
x55=5.03530051557472⋅10−17x_{55} = 5.03530051557472 \cdot 10^{-17}
x56=7.24511386253541⋅10−17x_{56} = 7.24511386253541 \cdot 10^{-17}
x57=2.41321271410711⋅10−15x_{57} = 2.41321271410711 \cdot 10^{-15}
x58=−1.11679598099599⋅10−15x_{58} = -1.11679598099599 \cdot 10^{-15}
x59=8.10018196837667⋅10−17x_{59} = 8.10018196837667 \cdot 10^{-17}
x60=−3.00353497873608⋅10−17x_{60} = -3.00353497873608 \cdot 10^{-17}
x61=−1.20485599377412⋅10−18x_{61} = -1.20485599377412 \cdot 10^{-18}
x62=2.22808074492689⋅10−16x_{62} = 2.22808074492689 \cdot 10^{-16}
x63=−3.25969474832924⋅10−14x_{63} = -3.25969474832924 \cdot 10^{-14}
x64=−1.76862305992214⋅10−14x_{64} = -1.76862305992214 \cdot 10^{-14}
x65=−2.80851332268296⋅10−16x_{65} = -2.80851332268296 \cdot 10^{-16}
x66=−6.9313016200319⋅10−18x_{66} = -6.9313016200319 \cdot 10^{-18}
x67=1.2355770545046⋅10−17x_{67} = 1.2355770545046 \cdot 10^{-17}
x68=7.15722835865887⋅10−18x_{68} = 7.15722835865887 \cdot 10^{-18}
x69=−9.58452216486739⋅10−17x_{69} = -9.58452216486739 \cdot 10^{-17}
x70=1.9689054691717⋅10−17x_{70} = 1.9689054691717 \cdot 10^{-17}
x71=1.75246451767813⋅10−15x_{71} = 1.75246451767813 \cdot 10^{-15}
x72=1.19114989391017⋅10−16x_{72} = 1.19114989391017 \cdot 10^{-16}
x73=8.15378087315969⋅10−18x_{73} = 8.15378087315969 \cdot 10^{-18}
x74=−2.04328546954794⋅10−16x_{74} = -2.04328546954794 \cdot 10^{-16}
x75=3.1976288208021⋅10−14x_{75} = 3.1976288208021 \cdot 10^{-14}
x76=8.39272508711364⋅10−16x_{76} = 8.39272508711364 \cdot 10^{-16}
x77=3.53383582269823⋅10−18x_{77} = 3.53383582269823 \cdot 10^{-18}
x78=−2.1294523165041⋅10−14x_{78} = -2.1294523165041 \cdot 10^{-14}
x79=2.07101215609992⋅10−17x_{79} = 2.07101215609992 \cdot 10^{-17}
x80=−6.28937394428614⋅10−17x_{80} = -6.28937394428614 \cdot 10^{-17}
x81=2.26018040886806⋅10−13x_{81} = 2.26018040886806 \cdot 10^{-13}
The values of the extrema at the points:
(1.5270790486031883e-17, 2)

(2.2357686539750427e-17, 2)

(1.5966825917964395e-16, 2)

(-1.1572558241835437e-16, 2)

(-1.0312577482214217e-16, 2)

(3.263846341877079e-16, 2)

(3.899196864772304e-15, 2)

(-1.140812093937536e-18, 2)

(-9.788569976980496e-17, 2)

(-5.930466559937818e-17, 2)

(1.722855144631456e-14, 2)

(2.730739202956583e-18, 2)

(-1.2787828527813194e-14, 2)

(-8.177041808539089e-16, 2)

(1.4234110508016475e-17, 2)

(1.1171972463513221e-17, 2)

(9.948461021168444e-18, 2)

(9.634038982734767e-17, 2)

(3.817857981276714e-15, 2)

(-3.8606849292934865e-16, 2)

(1.2584248545524538e-16, 2)

(-2.3951484385736968e-16, 2)

(-4.7927337973863105e-19, 2)

(2.0736465771844e-19, 2)

(-1.4736938967190768e-13, 2)

(-6.394965453984489e-18, 2)

(-5.1907868071214814e-17, 2)

(5.421891500262799e-17, 2)

(-3.844886044632263e-17, 2)

(2.1937150499575476e-15, 2)

(1.3655405659277714e-15, 2)

(6.1636687991621444e-18, 2)

(-1.9680036684828037e-15, 2)

(-1.0875607273668173e-15, 2)

(-2.6694558137739254e-17, 2)

(-1.4647907753690458e-18, 2)

(1.679373646631346e-15, 2)

(8.436980870534007e-15, 2)

(2.839867518750307e-15, 2)

(-5.055696003548158e-17, 2)

(-1.9863529512231008e-19, 2)

(-2.6416387835273495e-16, 2)

(1.9067701048957716e-17, 2)

(-1.3622296168976195e-19, 2)

(-1.114174063981536e-15, 2)

(4.5093268113141585e-14, 2)

(2.1641423377405037e-17, 2)

(-3.8011929444822644e-16, 2)

(6.551835922294143e-19, 2)

(8.703554944158104e-15, 2)

(1.0494919667469625e-16, 2)

(-1.3758305585179636e-14, 2)

(-1.1823647648488985e-15, 2)

(-3.8788971242916326e-15, 2)

(5.0353005155747165e-17, 2)

(7.245113862535405e-17, 2)

(2.4132127141071104e-15, 2)

(-1.1167959809959936e-15, 2)

(8.10018196837667e-17, 2)

(-3.003534978736078e-17, 2)

(-1.204855993774117e-18, 2)

(2.228080744926894e-16, 2)

(-3.2596947483292436e-14, 2)

(-1.7686230599221415e-14, 2)

(-2.8085133226829605e-16, 2)

(-6.9313016200318954e-18, 2)

(1.2355770545046002e-17, 2)

(7.157228358658869e-18, 2)

(-9.584522164867388e-17, 2)

(1.9689054691716976e-17, 2)

(1.752464517678129e-15, 2)

(1.1911498939101655e-16, 2)

(8.153780873159693e-18, 2)

(-2.0432854695479367e-16, 2)

(3.1976288208021023e-14, 2)

(8.392725087113635e-16, 2)

(3.533835822698227e-18, 2)

(-2.1294523165041048e-14, 2)

(2.0710121560999218e-17, 2)

(-6.289373944286139e-17, 2)

(2.2601804088680598e-13, 2)


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:
The function has no minima
Maxima of the function at points:
x81=1.52707904860319⋅10−17x_{81} = 1.52707904860319 \cdot 10^{-17}
x81=2.23576865397504⋅10−17x_{81} = 2.23576865397504 \cdot 10^{-17}
x81=1.59668259179644⋅10−16x_{81} = 1.59668259179644 \cdot 10^{-16}
x81=−1.15725582418354⋅10−16x_{81} = -1.15725582418354 \cdot 10^{-16}
x81=−1.03125774822142⋅10−16x_{81} = -1.03125774822142 \cdot 10^{-16}
x81=3.26384634187708⋅10−16x_{81} = 3.26384634187708 \cdot 10^{-16}
x81=3.8991968647723⋅10−15x_{81} = 3.8991968647723 \cdot 10^{-15}
x81=−1.14081209393754⋅10−18x_{81} = -1.14081209393754 \cdot 10^{-18}
x81=−9.7885699769805⋅10−17x_{81} = -9.7885699769805 \cdot 10^{-17}
x81=−5.93046655993782⋅10−17x_{81} = -5.93046655993782 \cdot 10^{-17}
x81=1.72285514463146⋅10−14x_{81} = 1.72285514463146 \cdot 10^{-14}
x81=2.73073920295658⋅10−18x_{81} = 2.73073920295658 \cdot 10^{-18}
x81=−1.27878285278132⋅10−14x_{81} = -1.27878285278132 \cdot 10^{-14}
x81=−8.17704180853909⋅10−16x_{81} = -8.17704180853909 \cdot 10^{-16}
x81=1.42341105080165⋅10−17x_{81} = 1.42341105080165 \cdot 10^{-17}
x81=1.11719724635132⋅10−17x_{81} = 1.11719724635132 \cdot 10^{-17}
x81=9.94846102116844⋅10−18x_{81} = 9.94846102116844 \cdot 10^{-18}
x81=9.63403898273477⋅10−17x_{81} = 9.63403898273477 \cdot 10^{-17}
x81=3.81785798127671⋅10−15x_{81} = 3.81785798127671 \cdot 10^{-15}
x81=−3.86068492929349⋅10−16x_{81} = -3.86068492929349 \cdot 10^{-16}
x81=1.25842485455245⋅10−16x_{81} = 1.25842485455245 \cdot 10^{-16}
x81=−2.3951484385737⋅10−16x_{81} = -2.3951484385737 \cdot 10^{-16}
x81=−4.79273379738631⋅10−19x_{81} = -4.79273379738631 \cdot 10^{-19}
x81=2.0736465771844⋅10−19x_{81} = 2.0736465771844 \cdot 10^{-19}
x81=−1.47369389671908⋅10−13x_{81} = -1.47369389671908 \cdot 10^{-13}
x81=−6.39496545398449⋅10−18x_{81} = -6.39496545398449 \cdot 10^{-18}
x81=−5.19078680712148⋅10−17x_{81} = -5.19078680712148 \cdot 10^{-17}
x81=5.4218915002628⋅10−17x_{81} = 5.4218915002628 \cdot 10^{-17}
x81=−3.84488604463226⋅10−17x_{81} = -3.84488604463226 \cdot 10^{-17}
x81=2.19371504995755⋅10−15x_{81} = 2.19371504995755 \cdot 10^{-15}
x81=1.36554056592777⋅10−15x_{81} = 1.36554056592777 \cdot 10^{-15}
x81=6.16366879916214⋅10−18x_{81} = 6.16366879916214 \cdot 10^{-18}
x81=−1.9680036684828⋅10−15x_{81} = -1.9680036684828 \cdot 10^{-15}
x81=−1.08756072736682⋅10−15x_{81} = -1.08756072736682 \cdot 10^{-15}
x81=−2.66945581377393⋅10−17x_{81} = -2.66945581377393 \cdot 10^{-17}
x81=−1.46479077536905⋅10−18x_{81} = -1.46479077536905 \cdot 10^{-18}
x81=1.67937364663135⋅10−15x_{81} = 1.67937364663135 \cdot 10^{-15}
x81=8.43698087053401⋅10−15x_{81} = 8.43698087053401 \cdot 10^{-15}
x81=2.83986751875031⋅10−15x_{81} = 2.83986751875031 \cdot 10^{-15}
x81=−5.05569600354816⋅10−17x_{81} = -5.05569600354816 \cdot 10^{-17}
x81=−1.9863529512231⋅10−19x_{81} = -1.9863529512231 \cdot 10^{-19}
x81=−2.64163878352735⋅10−16x_{81} = -2.64163878352735 \cdot 10^{-16}
x81=1.90677010489577⋅10−17x_{81} = 1.90677010489577 \cdot 10^{-17}
x81=−1.36222961689762⋅10−19x_{81} = -1.36222961689762 \cdot 10^{-19}
x81=−1.11417406398154⋅10−15x_{81} = -1.11417406398154 \cdot 10^{-15}
x81=4.50932681131416⋅10−14x_{81} = 4.50932681131416 \cdot 10^{-14}
x81=2.1641423377405⋅10−17x_{81} = 2.1641423377405 \cdot 10^{-17}
x81=−3.80119294448226⋅10−16x_{81} = -3.80119294448226 \cdot 10^{-16}
x81=6.55183592229414⋅10−19x_{81} = 6.55183592229414 \cdot 10^{-19}
x81=8.7035549441581⋅10−15x_{81} = 8.7035549441581 \cdot 10^{-15}
x81=1.04949196674696⋅10−16x_{81} = 1.04949196674696 \cdot 10^{-16}
x81=−1.37583055851796⋅10−14x_{81} = -1.37583055851796 \cdot 10^{-14}
x81=−1.1823647648489⋅10−15x_{81} = -1.1823647648489 \cdot 10^{-15}
x81=−3.87889712429163⋅10−15x_{81} = -3.87889712429163 \cdot 10^{-15}
x81=5.03530051557472⋅10−17x_{81} = 5.03530051557472 \cdot 10^{-17}
x81=7.24511386253541⋅10−17x_{81} = 7.24511386253541 \cdot 10^{-17}
x81=2.41321271410711⋅10−15x_{81} = 2.41321271410711 \cdot 10^{-15}
x81=−1.11679598099599⋅10−15x_{81} = -1.11679598099599 \cdot 10^{-15}
x81=8.10018196837667⋅10−17x_{81} = 8.10018196837667 \cdot 10^{-17}
x81=−3.00353497873608⋅10−17x_{81} = -3.00353497873608 \cdot 10^{-17}
x81=−1.20485599377412⋅10−18x_{81} = -1.20485599377412 \cdot 10^{-18}
x81=2.22808074492689⋅10−16x_{81} = 2.22808074492689 \cdot 10^{-16}
x81=−3.25969474832924⋅10−14x_{81} = -3.25969474832924 \cdot 10^{-14}
x81=−1.76862305992214⋅10−14x_{81} = -1.76862305992214 \cdot 10^{-14}
x81=−2.80851332268296⋅10−16x_{81} = -2.80851332268296 \cdot 10^{-16}
x81=−6.9313016200319⋅10−18x_{81} = -6.9313016200319 \cdot 10^{-18}
x81=1.2355770545046⋅10−17x_{81} = 1.2355770545046 \cdot 10^{-17}
x81=7.15722835865887⋅10−18x_{81} = 7.15722835865887 \cdot 10^{-18}
x81=−9.58452216486739⋅10−17x_{81} = -9.58452216486739 \cdot 10^{-17}
x81=1.9689054691717⋅10−17x_{81} = 1.9689054691717 \cdot 10^{-17}
x81=1.75246451767813⋅10−15x_{81} = 1.75246451767813 \cdot 10^{-15}
x81=1.19114989391017⋅10−16x_{81} = 1.19114989391017 \cdot 10^{-16}
x81=8.15378087315969⋅10−18x_{81} = 8.15378087315969 \cdot 10^{-18}
x81=−2.04328546954794⋅10−16x_{81} = -2.04328546954794 \cdot 10^{-16}
x81=3.1976288208021⋅10−14x_{81} = 3.1976288208021 \cdot 10^{-14}
x81=8.39272508711364⋅10−16x_{81} = 8.39272508711364 \cdot 10^{-16}
x81=3.53383582269823⋅10−18x_{81} = 3.53383582269823 \cdot 10^{-18}
x81=−2.1294523165041⋅10−14x_{81} = -2.1294523165041 \cdot 10^{-14}
x81=2.07101215609992⋅10−17x_{81} = 2.07101215609992 \cdot 10^{-17}
x81=−6.28937394428614⋅10−17x_{81} = -6.28937394428614 \cdot 10^{-17}
x81=2.26018040886806⋅10−13x_{81} = 2.26018040886806 \cdot 10^{-13}
Decreasing at intervals
(−∞,−1.47369389671908⋅10−13]\left(-\infty, -1.47369389671908 \cdot 10^{-13}\right]
Increasing at intervals
[2.26018040886806⋅10−13,∞)\left[2.26018040886806 \cdot 10^{-13}, \infty\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
x(cot⁡2(x2)+1)cot⁡(x2)2−cot⁡2(x2)−1=0\frac{x \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}}{2} - \cot^{2}{\left(\frac{x}{2} \right)} - 1 = 0
Solve this equation
The roots of this equation
x1=40.7426059185751x_{1} = 40.7426059185751
x2=−65.912778079645x_{2} = -65.912778079645
x3=21.8082433188578x_{3} = 21.8082433188578
x4=28.1323878256629x_{4} = 28.1323878256629
x5=−53.3321085176254x_{5} = -53.3321085176254
x6=−34.4415105438615x_{6} = -34.4415105438615
x7=84.7758271362638x_{7} = 84.7758271362638
x8=72.2012444887512x_{8} = 72.2012444887512
x9=53.3321085176254x_{9} = 53.3321085176254
x10=91.0622680279826x_{10} = 91.0622680279826
x11=−8.98681891581813x_{11} = -8.98681891581813
x12=−28.1323878256629x_{12} = -28.1323878256629
x13=−84.7758271362638x_{13} = -84.7758271362638
x14=−59.6231975817859x_{14} = -59.6231975817859
x15=−91.0622680279826x_{15} = -91.0622680279826
x16=47.038904997378x_{16} = 47.038904997378
x17=97.3482884639088x_{17} = 97.3482884639088
x18=−78.4888647223284x_{18} = -78.4888647223284
x19=34.4415105438615x_{19} = 34.4415105438615
x20=−21.8082433188578x_{20} = -21.8082433188578
x21=−72.2012444887512x_{21} = -72.2012444887512
x22=−97.3482884639088x_{22} = -97.3482884639088
x23=8.98681891581813x_{23} = 8.98681891581813
x24=78.4888647223284x_{24} = 78.4888647223284
x25=65.912778079645x_{25} = 65.912778079645
x26=−15.4505036738754x_{26} = -15.4505036738754
x27=59.6231975817859x_{27} = 59.6231975817859
x28=15.4505036738754x_{28} = 15.4505036738754
x29=−47.038904997378x_{29} = -47.038904997378
x30=−40.7426059185751x_{30} = -40.7426059185751

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
[−8.98681891581813,8.98681891581813]\left[-8.98681891581813, 8.98681891581813\right]
Convex at the intervals
(−∞,−97.3482884639088]\left(-\infty, -97.3482884639088\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(xcot⁡(x2))y = \lim_{x \to -\infty}\left(x \cot{\left(\frac{x}{2} \right)}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(xcot⁡(x2))y = \lim_{x \to \infty}\left(x \cot{\left(\frac{x}{2} \right)}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x*cot(x/2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞cot⁡(x2)=−cot⁡(∞)\lim_{x \to -\infty} \cot{\left(\frac{x}{2} \right)} = - \cot{\left(\infty \right)}
Let's take the limit
so,
inclined asymptote equation on the left:
y=−xcot⁡(∞)y = - x \cot{\left(\infty \right)}
lim⁡x→∞cot⁡(x2)=cot⁡(∞)\lim_{x \to \infty} \cot{\left(\frac{x}{2} \right)} = \cot{\left(\infty \right)}
Let's take the limit
so,
inclined asymptote equation on the right:
y=xcot⁡(∞)y = x \cot{\left(\infty \right)}
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:
xcot⁡(x2)=xcot⁡(x2)x \cot{\left(\frac{x}{2} \right)} = x \cot{\left(\frac{x}{2} \right)}
- No
xcot⁡(x2)=−xcot⁡(x2)x \cot{\left(\frac{x}{2} \right)} = - x \cot{\left(\frac{x}{2} \right)}
- No
so, the function
not is
neither even, nor odd
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