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Plotting a function graph/ 2-x*arcctg(x)

Graphing y = 2-x*arcctg(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = 2 - x*acot(x)
f(x)=−xacot⁡(x)+2f{\left(x \right)} = - x \operatorname{acot}{\left(x \right)} + 2 
f = -x*acot(x) + 2
The graph of the function
02468-8-6-4-2-101002
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)+2=0- x \operatorname{acot}{\left(x \right)} + 2 = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2 - x*acot(x).
−0acot⁡(0)+2- 0 \operatorname{acot}{\left(0 \right)} + 2
The result:
f(0)=2f{\left(0 \right)} = 2
The point:
(0, 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
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=−36383.6664480288x_{1} = -36383.6664480288
x2=30582.10639855x_{2} = 30582.10639855
x3=−16891.4245591221x_{3} = -16891.4245591221
x4=33972.2501557217x_{4} = 33972.2501557217
x5=19564.7930550097x_{5} = 19564.7930550097
x6=8551.45303022722x_{6} = 8551.45303022722
x7=42447.8004663952x_{7} = 42447.8004663952
x8=28887.0585662606x_{8} = 28887.0585662606
x9=39905.113426091x_{9} = 39905.113426091
x10=−32993.4859340089x_{10} = -32993.4859340089
x11=35667.3415095204x_{11} = 35667.3415095204
x12=40752.6740304247x_{12} = 40752.6740304247
x13=−35536.1170902848x_{13} = -35536.1170902848
x14=−34688.5704142599x_{14} = -34688.5704142599
x15=−17738.7919818582x_{15} = -17738.7919818582
x16=31429.6368125843x_{16} = 31429.6368125843
x17=−25365.810848258x_{17} = -25365.810848258
x18=24649.5356458528x_{18} = 24649.5356458528
x19=16175.2730314267x_{19} = 16175.2730314267
x20=−38078.7724924216x_{20} = -38078.7724924216
x21=39057.5547154358x_{21} = 39057.5547154358
x22=−15196.7689456217x_{22} = -15196.7689456217
x23=34819.7944072015x_{23} = 34819.7944072015
x24=−19433.5887869564x_{24} = -19433.5887869564
x25=−39773.8872675524x_{25} = -39773.8872675524
x26=−21975.9030580391x_{26} = -21975.9030580391
x27=−10960.8778764425x_{27} = -10960.8778764425
x28=−38926.3288599295x_{28} = -38926.3288599295
x29=20412.2196081348x_{29} = 20412.2196081348
x30=36514.8912641685x_{30} = 36514.8912641685
x31=26344.5255783821x_{31} = 26344.5255783821
x32=22954.5779890773x_{32} = 22954.5779890773
x33=−22823.365825763x_{33} = -22823.365825763
x34=14480.6704897809x_{34} = 14480.6704897809
x35=38209.9980244722x_{35} = 38209.9980244722
x36=−27060.8125814987x_{36} = -27060.8125814987
x37=−20281.012992025x_{37} = -20281.012992025
x38=23802.0523520614x_{38} = 23802.0523520614
x39=−14349.4901108771x_{39} = -14349.4901108771
x40=−12655.0631680385x_{40} = -12655.0631680385
x41=−31298.4148843908x_{41} = -31298.4148843908
x42=−24518.3207010738x_{42} = -24518.3207010738
x43=11092.020896022x_{43} = 11092.020896022
x44=25497.0269799384x_{44} = 25497.0269799384
x45=−29603.3595724168x_{45} = -29603.3595724168
x46=−37231.2183043527x_{46} = -37231.2183043527
x47=15327.9550113123x_{47} = 15327.9550113123
x48=11939.0892257274x_{48} = 11939.0892257274
x49=−21128.451639271x_{49} = -21128.451639271
x50=28039.5419336107x_{50} = 28039.5419336107
x51=−30450.8850969459x_{51} = -30450.8850969459
x52=17869.99050832x_{52} = 17869.99050832
x53=12786.2285767811x_{53} = 12786.2285767811
x54=−27908.3228655288x_{54} = -27908.3228655288
x55=33124.7089739066x_{55} = 33124.7089739066
x56=37362.4434906472x_{56} = 37362.4434906472
x57=−23670.8387235676x_{57} = -23670.8387235676
x58=−9267.06673722176x_{58} = -9267.06673722176
x59=−28755.8386877142x_{59} = -28755.8386877142
x60=−8420.37201352502x_{60} = -8420.37201352502
x61=41600.2364126928x_{61} = 41600.2364126928
x62=27192.0307619732x_{62} = 27192.0307619732
x63=−42316.5735052903x_{63} = -42316.5735052903
x64=−18586.1809993501x_{64} = -18586.1809993501
x65=−26213.3083727677x_{65} = -26213.3083727677
x66=−13502.2521178288x_{66} = -13502.2521178288
x67=13633.4257117535x_{67} = 13633.4257117535
x68=−33841.0266214391x_{68} = -33841.0266214391
x69=−10113.913790961x_{69} = -10113.913790961
x70=−40621.447587593x_{70} = -40621.447587593
x71=−16044.0821523617x_{71} = -16044.0821523617
x72=32277.1711035644x_{72} = 32277.1711035644
x73=17022.6195482377x_{73} = 17022.6195482377
x74=18717.3825921039x_{74} = 18717.3825921039
x75=−41469.0097027937x_{75} = -41469.0097027937
x76=22107.1135839482x_{76} = 22107.1135839482
x77=21259.6603274632x_{77} = 21259.6603274632
x78=−11807.9338157736x_{78} = -11807.9338157736
x79=29734.5801929714x_{79} = 29734.5801929714
x80=−32145.9485975863x_{80} = -32145.9485975863
x81=10245.0412006996x_{81} = 10245.0412006996
x82=9398.17409812154x_{82} = 9398.17409812154
The values of the extrema at the points:
(-36383.66644802879, 1.00000000025181)

(30582.106398549957, 1.00000000035641)

(-16891.424559122082, 1.00000000116828)

(33972.25015572168, 1.00000000028882)

(19564.79305500968, 1.00000000087082)

(8551.453030227216, 1.00000000455826)

(42447.8004663952, 1.000000000185)

(28887.058566260628, 1.00000000039946)

(39905.113426091, 1.00000000020933)

(-32993.48593400886, 1.00000000030621)

(35667.3415095204, 1.00000000026202)

(40752.67403042468, 1.00000000020071)

(-35536.11709028476, 1.00000000026396)

(-34688.570414259906, 1.00000000027702)

(-17738.791981858223, 1.00000000105933)

(31429.636812584253, 1.00000000033744)

(-25365.810848258036, 1.00000000051806)

(24649.53564585282, 1.00000000054861)

(16175.273031426721, 1.00000000127402)

(-38078.77249242158, 1.00000000022989)

(39057.55471543576, 1.00000000021851)

(-15196.768945621745, 1.00000000144336)

(34819.79440720148, 1.00000000027493)

(-19433.58878695641, 1.00000000088262)

(-39773.887267552374, 1.00000000021071)

(-21975.903058039054, 1.00000000069022)

(-10960.87787644252, 1.00000000277452)

(-38926.32885992955, 1.00000000021998)

(20412.219608134823, 1.00000000080002)

(36514.89126416845, 1.00000000025)

(26344.525578382065, 1.00000000048028)

(22954.57798907731, 1.00000000063262)

(-22823.365825763038, 1.00000000063991)

(14480.670489780925, 1.00000000158965)

(38209.99802447215, 1.00000000022831)

(-27060.81258149867, 1.00000000045519)

(-20281.012992024975, 1.0000000008104)

(23802.052352061375, 1.00000000058837)

(-14349.490110877094, 1.00000000161885)

(-12655.063168038458, 1.00000000208137)

(-31298.41488439077, 1.00000000034028)

(-24518.320701073833, 1.00000000055449)

(11092.020896021957, 1.0000000027093)

(25497.02697993837, 1.00000000051274)

(-29603.35957241684, 1.00000000038036)

(-37231.21830435275, 1.00000000024047)

(15327.955011312279, 1.00000000141876)

(11939.089225727435, 1.00000000233849)

(-21128.451639271025, 1.0000000007467)

(28039.541933610733, 1.00000000042397)

(-30450.88509694595, 1.00000000035948)

(17869.990508320032, 1.00000000104383)

(12786.228576781117, 1.00000000203889)

(-27908.322865528775, 1.00000000042797)

(33124.7089739066, 1.00000000030379)

(37362.44349064715, 1.00000000023879)

(-23670.838723567595, 1.00000000059491)

(-9267.06673722176, 1.00000000388145)

(-28755.838687714244, 1.00000000040311)

(-8420.37201352502, 1.00000000470128)

(41600.23641269277, 1.00000000019261)

(27192.03076197315, 1.00000000045081)

(-42316.573505290275, 1.00000000018615)

(-18586.180999350086, 1.00000000096494)

(-26213.308372767653, 1.0000000004851)

(-13502.25211782884, 1.00000000182838)

(13633.425711753476, 1.00000000179337)

(-33841.02662143908, 1.00000000029107)

(-10113.913790961042, 1.00000000325867)

(-40621.447587593044, 1.00000000020201)

(-16044.082152361707, 1.00000000129494)

(32277.171103564393, 1.00000000031995)

(17022.619548237748, 1.00000000115034)

(18717.38259210387, 1.00000000095146)

(-41469.00970279369, 1.00000000019383)

(22107.113583948245, 1.00000000068205)

(21259.66032746322, 1.00000000073751)

(-11807.933815773578, 1.00000000239073)

(29734.580192971367, 1.00000000037701)

(-32145.948597586266, 1.00000000032257)

(10245.041200699563, 1.00000000317579)

(9398.174098121537, 1.00000000377391)


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=39057.5547154358x_{1} = 39057.5547154358
x2=34819.7944072015x_{2} = 34819.7944072015
x3=−37231.2183043527x_{3} = -37231.2183043527
x4=11939.0892257274x_{4} = 11939.0892257274
x5=37362.4434906472x_{5} = 37362.4434906472
x6=18717.3825921039x_{6} = 18717.3825921039
x7=22107.1135839482x_{7} = 22107.1135839482
Maxima of the function at points:
x7=−32993.4859340089x_{7} = -32993.4859340089
x7=31429.6368125843x_{7} = 31429.6368125843
x7=−38078.7724924216x_{7} = -38078.7724924216
x7=14480.6704897809x_{7} = 14480.6704897809
x7=−14349.4901108771x_{7} = -14349.4901108771
x7=25497.0269799384x_{7} = 25497.0269799384
x7=−42316.5735052903x_{7} = -42316.5735052903
x7=−26213.3083727677x_{7} = -26213.3083727677
Decreasing at intervals
[39057.5547154358,∞)\left[39057.5547154358, \infty\right)
Increasing at intervals
(−∞,−37231.2183043527]\left(-\infty, -37231.2183043527\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)+2)=1\lim_{x \to -\infty}\left(- x \operatorname{acot}{\left(x \right)} + 2\right) = 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=1y = 1
lim⁡x→∞(−xacot⁡(x)+2)=1\lim_{x \to \infty}\left(- x \operatorname{acot}{\left(x \right)} + 2\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 2 - x*acot(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−xacot⁡(x)+2x)=0\lim_{x \to -\infty}\left(\frac{- x \operatorname{acot}{\left(x \right)} + 2}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−xacot⁡(x)+2x)=0\lim_{x \to \infty}\left(\frac{- x \operatorname{acot}{\left(x \right)} + 2}{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)+2=−xacot⁡(x)+2- x \operatorname{acot}{\left(x \right)} + 2 = - x \operatorname{acot}{\left(x \right)} + 2
- Yes
−xacot⁡(x)+2=xacot⁡(x)−2- x \operatorname{acot}{\left(x \right)} + 2 = x \operatorname{acot}{\left(x \right)} - 2
- No
so, the function
is
even
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