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Plotting a function graph/ x*arcctg(x)

Graphing y = x*arcctg(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = x*acot(x)
f(x)=xacot⁡(x)f{\left(x \right)} = x \operatorname{acot}{\left(x \right)} 
f = x*acot(x)
The graph of the function
0-80-60-40-2020406080-10010002
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:
xacot⁡(x)=0x \operatorname{acot}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*acot(x).
0acot⁡(0)0 \operatorname{acot}{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
−xx2+1+acot⁡(x)=0- \frac{x}{x^{2} + 1} + \operatorname{acot}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−34688.5704142599x_{1} = -34688.5704142599
x2=33972.2501557217x_{2} = 33972.2501557217
x3=−27060.8125814987x_{3} = -27060.8125814987
x4=−20281.012992025x_{4} = -20281.012992025
x5=22107.1135839482x_{5} = 22107.1135839482
x6=−30450.8850969459x_{6} = -30450.8850969459
x7=15327.9550113123x_{7} = 15327.9550113123
x8=39057.5547154358x_{8} = 39057.5547154358
x9=−16044.0821523617x_{9} = -16044.0821523617
x10=−10113.913790961x_{10} = -10113.913790961
x11=−42316.5735052903x_{11} = -42316.5735052903
x12=−36383.6664480288x_{12} = -36383.6664480288
x13=−24518.3207010738x_{13} = -24518.3207010738
x14=−35536.1170902848x_{14} = -35536.1170902848
x15=−12655.0631680385x_{15} = -12655.0631680385
x16=37362.4434906472x_{16} = 37362.4434906472
x17=−8420.37201352502x_{17} = -8420.37201352502
x18=8551.45303022722x_{18} = 8551.45303022722
x19=38209.9980244722x_{19} = 38209.9980244722
x20=11092.020896022x_{20} = 11092.020896022
x21=−32145.9485975863x_{21} = -32145.9485975863
x22=27192.0307619732x_{22} = 27192.0307619732
x23=13633.4257117535x_{23} = 13633.4257117535
x24=−41469.0097027937x_{24} = -41469.0097027937
x25=10245.0412006996x_{25} = 10245.0412006996
x26=28039.5419336107x_{26} = 28039.5419336107
x27=22954.5779890773x_{27} = 22954.5779890773
x28=28887.0585662606x_{28} = 28887.0585662606
x29=−18586.1809993501x_{29} = -18586.1809993501
x30=35667.3415095204x_{30} = 35667.3415095204
x31=−16891.4245591221x_{31} = -16891.4245591221
x32=−21975.9030580391x_{32} = -21975.9030580391
x33=−9267.06673722176x_{33} = -9267.06673722176
x34=−11807.9338157736x_{34} = -11807.9338157736
x35=12786.2285767811x_{35} = 12786.2285767811
x36=42447.8004663952x_{36} = 42447.8004663952
x37=16175.2730314267x_{37} = 16175.2730314267
x38=−23670.8387235676x_{38} = -23670.8387235676
x39=23802.0523520614x_{39} = 23802.0523520614
x40=−26213.3083727677x_{40} = -26213.3083727677
x41=−19433.5887869564x_{41} = -19433.5887869564
x42=−27908.3228655288x_{42} = -27908.3228655288
x43=−39773.8872675524x_{43} = -39773.8872675524
x44=17869.99050832x_{44} = 17869.99050832
x45=−32993.4859340089x_{45} = -32993.4859340089
x46=−38926.3288599295x_{46} = -38926.3288599295
x47=−22823.365825763x_{47} = -22823.365825763
x48=−17738.7919818582x_{48} = -17738.7919818582
x49=33124.7089739066x_{49} = 33124.7089739066
x50=19564.7930550097x_{50} = 19564.7930550097
x51=26344.5255783821x_{51} = 26344.5255783821
x52=31429.6368125843x_{52} = 31429.6368125843
x53=14480.6704897809x_{53} = 14480.6704897809
x54=36514.8912641685x_{54} = 36514.8912641685
x55=34819.7944072015x_{55} = 34819.7944072015
x56=−31298.4148843908x_{56} = -31298.4148843908
x57=−10960.8778764425x_{57} = -10960.8778764425
x58=−40621.447587593x_{58} = -40621.447587593
x59=18717.3825921039x_{59} = 18717.3825921039
x60=41600.2364126928x_{60} = 41600.2364126928
x61=24649.5356458528x_{61} = 24649.5356458528
x62=−28755.8386877142x_{62} = -28755.8386877142
x63=32277.1711035644x_{63} = 32277.1711035644
x64=−33841.0266214391x_{64} = -33841.0266214391
x65=40752.6740304247x_{65} = 40752.6740304247
x66=−25365.810848258x_{66} = -25365.810848258
x67=17022.6195482377x_{67} = 17022.6195482377
x68=−13502.2521178288x_{68} = -13502.2521178288
x69=−21128.451639271x_{69} = -21128.451639271
x70=9398.17409812154x_{70} = 9398.17409812154
x71=20412.2196081348x_{71} = 20412.2196081348
x72=29734.5801929714x_{72} = 29734.5801929714
x73=25497.0269799384x_{73} = 25497.0269799384
x74=−14349.4901108771x_{74} = -14349.4901108771
x75=−38078.7724924216x_{75} = -38078.7724924216
x76=11939.0892257274x_{76} = 11939.0892257274
x77=−15196.7689456217x_{77} = -15196.7689456217
x78=−29603.3595724168x_{78} = -29603.3595724168
x79=21259.6603274632x_{79} = 21259.6603274632
x80=−37231.2183043527x_{80} = -37231.2183043527
x81=30582.10639855x_{81} = 30582.10639855
x82=39905.113426091x_{82} = 39905.113426091
The values of the extrema at the points:
(-34688.570414259906, 0.999999999722983)

(33972.25015572168, 0.999999999711178)

(-27060.81258149867, 0.999999999544805)

(-20281.012992024975, 0.9999999991896)

(22107.113583948245, 0.999999999317952)

(-30450.88509694595, 0.999999999640517)

(15327.955011312279, 0.999999998581235)

(39057.55471543576, 0.999999999781491)

(-16044.082152361707, 0.999999998705062)

(-10113.913790961042, 0.999999996741331)

(-42316.573505290275, 0.999999999813852)

(-36383.66644802879, 0.999999999748194)

(-24518.320701073833, 0.999999999445505)

(-35536.11709028476, 0.99999999973604)

(-12655.063168038458, 0.999999997918626)

(37362.44349064715, 0.999999999761214)

(-8420.37201352502, 0.999999995298719)

(8551.453030227216, 0.999999995441742)

(38209.99802447215, 0.99999999977169)

(11092.020896021957, 0.999999997290698)

(-32145.948597586266, 0.999999999677428)

(27192.03076197315, 0.999999999549188)

(13633.425711753476, 0.999999998206635)

(-41469.00970279369, 0.999999999806165)

(10245.041200699563, 0.999999996824213)

(28039.541933610733, 0.999999999576028)

(22954.57798907731, 0.999999999367384)

(28887.058566260628, 0.999999999600541)

(-18586.180999350086, 0.999999999035064)

(35667.3415095204, 0.999999999737978)

(-16891.424559122082, 0.999999998831722)

(-21975.903058039054, 0.999999999309784)

(-9267.06673722176, 0.999999996118549)

(-11807.933815773578, 0.999999997609268)

(12786.228576781117, 0.99999999796111)

(42447.8004663952, 0.999999999815001)

(16175.273031426721, 0.999999998725982)

(-23670.838723567595, 0.99999999940509)

(23802.052352061375, 0.999999999411631)

(-26213.308372767653, 0.999999999514896)

(-19433.58878695641, 0.999999999117382)

(-27908.322865528775, 0.999999999572032)

(-39773.887267552374, 0.999999999789291)

(17869.990508320032, 0.999999998956169)

(-32993.48593400886, 0.999999999693788)

(-38926.32885992955, 0.999999999780016)

(-22823.365825763038, 0.999999999360089)

(-17738.791981858223, 0.999999998940671)

(33124.7089739066, 0.999999999696209)

(19564.79305500968, 0.99999999912918)

(26344.525578382065, 0.999999999519716)

(31429.636812584253, 0.999999999662557)

(14480.670489780925, 0.99999999841035)

(36514.89126416845, 0.999999999750001)

(34819.79440720148, 0.999999999725067)

(-31298.41488439077, 0.999999999659722)

(-10960.87787644252, 0.999999997225479)

(-40621.447587593044, 0.999999999797992)

(18717.38259210387, 0.999999999048545)

(41600.23641269277, 0.999999999807386)

(24649.53564585282, 0.999999999451393)

(-28755.838687714244, 0.999999999596887)

(32277.171103564393, 0.999999999680046)

(-33841.02662143908, 0.999999999708934)

(40752.67403042468, 0.999999999799291)

(-25365.810848258036, 0.999999999481939)

(17022.619548237748, 0.999999998849661)

(-13502.25211782884, 0.999999998171621)

(-21128.451639271025, 0.999999999253305)

(9398.174098121537, 0.999999996226088)

(20412.219608134823, 0.999999999199985)

(29734.580192971367, 0.999999999622988)

(25497.02697993837, 0.999999999487257)

(-14349.490110877094, 0.999999998381153)

(-38078.77249242158, 0.999999999770114)

(11939.089225727435, 0.999999997661506)

(-15196.768945621745, 0.999999998556635)

(-29603.35957241684, 0.999999999619638)

(21259.66032746322, 0.999999999262493)

(-37231.21830435275, 0.999999999759528)

(30582.106398549957, 0.999999999643595)

(39905.113426091, 0.999999999790675)


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=−42316.5735052903x_{1} = -42316.5735052903
x2=−26213.3083727677x_{2} = -26213.3083727677
x3=−32993.4859340089x_{3} = -32993.4859340089
x4=−22823.365825763x_{4} = -22823.365825763
x5=31429.6368125843x_{5} = 31429.6368125843
x6=14480.6704897809x_{6} = 14480.6704897809
x7=−40621.447587593x_{7} = -40621.447587593
x8=24649.5356458528x_{8} = 24649.5356458528
x9=32277.1711035644x_{9} = 32277.1711035644
x10=40752.6740304247x_{10} = 40752.6740304247
x11=17022.6195482377x_{11} = 17022.6195482377
x12=25497.0269799384x_{12} = 25497.0269799384
x13=−14349.4901108771x_{13} = -14349.4901108771
x14=−38078.7724924216x_{14} = -38078.7724924216
Maxima of the function at points:
x14=22107.1135839482x_{14} = 22107.1135839482
x14=39057.5547154358x_{14} = 39057.5547154358
x14=−16044.0821523617x_{14} = -16044.0821523617
x14=37362.4434906472x_{14} = 37362.4434906472
x14=38209.9980244722x_{14} = 38209.9980244722
x14=−32145.9485975863x_{14} = -32145.9485975863
x14=10245.0412006996x_{14} = 10245.0412006996
x14=−16891.4245591221x_{14} = -16891.4245591221
x14=34819.7944072015x_{14} = 34819.7944072015
x14=18717.3825921039x_{14} = 18717.3825921039
x14=11939.0892257274x_{14} = 11939.0892257274
x14=−37231.2183043527x_{14} = -37231.2183043527
Decreasing at intervals
[40752.6740304247,∞)\left[40752.6740304247, \infty\right)
Increasing at intervals
(−∞,−42316.5735052903]\left(-\infty, -42316.5735052903\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
2(x2x2+1−1)x2+1=0\frac{2 \left(\frac{x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xacot⁡(x))=1\lim_{x \to -\infty}\left(x \operatorname{acot}{\left(x \right)}\right) = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡x→∞(xacot⁡(x))=1\lim_{x \to \infty}\left(x \operatorname{acot}{\left(x \right)}\right) = 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=1y = 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x*acot(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞acot⁡(x)=0\lim_{x \to -\infty} \operatorname{acot}{\left(x \right)} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞acot⁡(x)=0\lim_{x \to \infty} \operatorname{acot}{\left(x \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:
xacot⁡(x)=xacot⁡(x)x \operatorname{acot}{\left(x \right)} = x \operatorname{acot}{\left(x \right)}
- Yes
xacot⁡(x)=−xacot⁡(x)x \operatorname{acot}{\left(x \right)} = - x \operatorname{acot}{\left(x \right)}
- No
so, the function
is
even
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