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xarcctg(x)
Plotting a function graph/ xarcctg(x)

Graphing y = xarcctg(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
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)=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
acot⁡(x)−xx2+1=0\operatorname{acot}{\left(x \right)} - \frac{x}{x^{2} + 1} = 0
Solve this equation
The roots of this equation
x1=−40621.447587593x_{1} = -40621.447587593
x2=−31298.4148843908x_{2} = -31298.4148843908
x3=−34688.5704142599x_{3} = -34688.5704142599
x4=−30450.8850969459x_{4} = -30450.8850969459
x5=−38926.3288599295x_{5} = -38926.3288599295
x6=16175.2730314267x_{6} = 16175.2730314267
x7=30582.10639855x_{7} = 30582.10639855
x8=33972.2501557217x_{8} = 33972.2501557217
x9=17022.6195482377x_{9} = 17022.6195482377
x10=−21128.451639271x_{10} = -21128.451639271
x11=39905.113426091x_{11} = 39905.113426091
x12=−27908.3228655288x_{12} = -27908.3228655288
x13=−19433.5887869564x_{13} = -19433.5887869564
x14=33124.7089739066x_{14} = 33124.7089739066
x15=−16891.4245591221x_{15} = -16891.4245591221
x16=22107.1135839482x_{16} = 22107.1135839482
x17=−13502.2521178288x_{17} = -13502.2521178288
x18=17869.99050832x_{18} = 17869.99050832
x19=−33841.0266214391x_{19} = -33841.0266214391
x20=36514.8912641685x_{20} = 36514.8912641685
x21=9398.17409812154x_{21} = 9398.17409812154
x22=29734.5801929714x_{22} = 29734.5801929714
x23=−32145.9485975863x_{23} = -32145.9485975863
x24=8551.45303022722x_{24} = 8551.45303022722
x25=−17738.7919818582x_{25} = -17738.7919818582
x26=−29603.3595724168x_{26} = -29603.3595724168
x27=−11807.9338157736x_{27} = -11807.9338157736
x28=−35536.1170902848x_{28} = -35536.1170902848
x29=−32993.4859340089x_{29} = -32993.4859340089
x30=38209.9980244722x_{30} = 38209.9980244722
x31=−42316.5735052903x_{31} = -42316.5735052903
x32=19564.7930550097x_{32} = 19564.7930550097
x33=−22823.365825763x_{33} = -22823.365825763
x34=40752.6740304247x_{34} = 40752.6740304247
x35=−20281.012992025x_{35} = -20281.012992025
x36=−37231.2183043527x_{36} = -37231.2183043527
x37=−10960.8778764425x_{37} = -10960.8778764425
x38=−18586.1809993501x_{38} = -18586.1809993501
x39=14480.6704897809x_{39} = 14480.6704897809
x40=11939.0892257274x_{40} = 11939.0892257274
x41=−24518.3207010738x_{41} = -24518.3207010738
x42=15327.9550113123x_{42} = 15327.9550113123
x43=10245.0412006996x_{43} = 10245.0412006996
x44=26344.5255783821x_{44} = 26344.5255783821
x45=−16044.0821523617x_{45} = -16044.0821523617
x46=−14349.4901108771x_{46} = -14349.4901108771
x47=37362.4434906472x_{47} = 37362.4434906472
x48=−26213.3083727677x_{48} = -26213.3083727677
x49=−25365.810848258x_{49} = -25365.810848258
x50=27192.0307619732x_{50} = 27192.0307619732
x51=18717.3825921039x_{51} = 18717.3825921039
x52=−15196.7689456217x_{52} = -15196.7689456217
x53=28039.5419336107x_{53} = 28039.5419336107
x54=−21975.9030580391x_{54} = -21975.9030580391
x55=24649.5356458528x_{55} = 24649.5356458528
x56=−39773.8872675524x_{56} = -39773.8872675524
x57=21259.6603274632x_{57} = 21259.6603274632
x58=−23670.8387235676x_{58} = -23670.8387235676
x59=42447.8004663952x_{59} = 42447.8004663952
x60=34819.7944072015x_{60} = 34819.7944072015
x61=−41469.0097027937x_{61} = -41469.0097027937
x62=−36383.6664480288x_{62} = -36383.6664480288
x63=41600.2364126928x_{63} = 41600.2364126928
x64=20412.2196081348x_{64} = 20412.2196081348
x65=35667.3415095204x_{65} = 35667.3415095204
x66=−28755.8386877142x_{66} = -28755.8386877142
x67=−9267.06673722176x_{67} = -9267.06673722176
x68=−38078.7724924216x_{68} = -38078.7724924216
x69=32277.1711035644x_{69} = 32277.1711035644
x70=−8420.37201352502x_{70} = -8420.37201352502
x71=11092.020896022x_{71} = 11092.020896022
x72=22954.5779890773x_{72} = 22954.5779890773
x73=25497.0269799384x_{73} = 25497.0269799384
x74=−12655.0631680385x_{74} = -12655.0631680385
x75=23802.0523520614x_{75} = 23802.0523520614
x76=39057.5547154358x_{76} = 39057.5547154358
x77=28887.0585662606x_{77} = 28887.0585662606
x78=12786.2285767811x_{78} = 12786.2285767811
x79=−27060.8125814987x_{79} = -27060.8125814987
x80=31429.6368125843x_{80} = 31429.6368125843
x81=13633.4257117535x_{81} = 13633.4257117535
x82=−10113.913790961x_{82} = -10113.913790961
The values of the extrema at the points:
(-40621.447587593, 0.999999999797992)

(-31298.4148843908, 0.999999999659722)

(-34688.5704142599, 0.999999999722983)

(-30450.8850969459, 0.999999999640517)

(-38926.3288599295, 0.999999999780016)

(16175.2730314267, 0.999999998725982)

(30582.10639855, 0.999999999643595)

(33972.2501557217, 0.999999999711178)

(17022.6195482377, 0.999999998849661)

(-21128.451639271, 0.999999999253305)

(39905.113426091, 0.999999999790675)

(-27908.3228655288, 0.999999999572032)

(-19433.5887869564, 0.999999999117382)

(33124.7089739066, 0.999999999696209)

(-16891.4245591221, 0.999999998831722)

(22107.1135839482, 0.999999999317952)

(-13502.2521178288, 0.999999998171621)

(17869.99050832, 0.999999998956169)

(-33841.0266214391, 0.999999999708934)

(36514.8912641685, 0.999999999750001)

(9398.17409812154, 0.999999996226088)

(29734.5801929714, 0.999999999622988)

(-32145.9485975863, 0.999999999677428)

(8551.45303022722, 0.999999995441742)

(-17738.7919818582, 0.999999998940671)

(-29603.3595724168, 0.999999999619638)

(-11807.9338157736, 0.999999997609268)

(-35536.1170902848, 0.99999999973604)

(-32993.4859340089, 0.999999999693788)

(38209.9980244722, 0.99999999977169)

(-42316.5735052903, 0.999999999813852)

(19564.7930550097, 0.99999999912918)

(-22823.365825763, 0.999999999360089)

(40752.6740304247, 0.999999999799291)

(-20281.012992025, 0.9999999991896)

(-37231.2183043527, 0.999999999759528)

(-10960.8778764425, 0.999999997225479)

(-18586.1809993501, 0.999999999035064)

(14480.6704897809, 0.99999999841035)

(11939.0892257274, 0.999999997661506)

(-24518.3207010738, 0.999999999445505)

(15327.9550113123, 0.999999998581235)

(10245.0412006996, 0.999999996824213)

(26344.5255783821, 0.999999999519716)

(-16044.0821523617, 0.999999998705062)

(-14349.4901108771, 0.999999998381153)

(37362.4434906472, 0.999999999761214)

(-26213.3083727677, 0.999999999514896)

(-25365.810848258, 0.999999999481939)

(27192.0307619732, 0.999999999549188)

(18717.3825921039, 0.999999999048545)

(-15196.7689456217, 0.999999998556635)

(28039.5419336107, 0.999999999576028)

(-21975.9030580391, 0.999999999309784)

(24649.5356458528, 0.999999999451393)

(-39773.8872675524, 0.999999999789291)

(21259.6603274632, 0.999999999262493)

(-23670.8387235676, 0.99999999940509)

(42447.8004663952, 0.999999999815001)

(34819.7944072015, 0.999999999725067)

(-41469.0097027937, 0.999999999806165)

(-36383.6664480288, 0.999999999748194)

(41600.2364126928, 0.999999999807386)

(20412.2196081348, 0.999999999199985)

(35667.3415095204, 0.999999999737978)

(-28755.8386877142, 0.999999999596887)

(-9267.06673722176, 0.999999996118549)

(-38078.7724924216, 0.999999999770114)

(32277.1711035644, 0.999999999680046)

(-8420.37201352502, 0.999999995298719)

(11092.020896022, 0.999999997290698)

(22954.5779890773, 0.999999999367384)

(25497.0269799384, 0.999999999487257)

(-12655.0631680385, 0.999999997918626)

(23802.0523520614, 0.999999999411631)

(39057.5547154358, 0.999999999781491)

(28887.0585662606, 0.999999999600541)

(12786.2285767811, 0.99999999796111)

(-27060.8125814987, 0.999999999544805)

(31429.6368125843, 0.999999999662557)

(13633.4257117535, 0.999999998206635)

(-10113.913790961, 0.999999996741331)


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=−40621.447587593x_{1} = -40621.447587593
x2=17022.6195482377x_{2} = 17022.6195482377
x3=−32993.4859340089x_{3} = -32993.4859340089
x4=−42316.5735052903x_{4} = -42316.5735052903
x5=−22823.365825763x_{5} = -22823.365825763
x6=40752.6740304247x_{6} = 40752.6740304247
x7=14480.6704897809x_{7} = 14480.6704897809
x8=−14349.4901108771x_{8} = -14349.4901108771
x9=−26213.3083727677x_{9} = -26213.3083727677
x10=24649.5356458528x_{10} = 24649.5356458528
x11=−38078.7724924216x_{11} = -38078.7724924216
x12=32277.1711035644x_{12} = 32277.1711035644
x13=25497.0269799384x_{13} = 25497.0269799384
x14=31429.6368125843x_{14} = 31429.6368125843
Maxima of the function at points:
x14=−16891.4245591221x_{14} = -16891.4245591221
x14=22107.1135839482x_{14} = 22107.1135839482
x14=−32145.9485975863x_{14} = -32145.9485975863
x14=38209.9980244722x_{14} = 38209.9980244722
x14=−37231.2183043527x_{14} = -37231.2183043527
x14=11939.0892257274x_{14} = 11939.0892257274
x14=10245.0412006996x_{14} = 10245.0412006996
x14=−16044.0821523617x_{14} = -16044.0821523617
x14=18717.3825921039x_{14} = 18717.3825921039
x14=34819.7944072015x_{14} = 34819.7944072015
x14=39057.5547154358x_{14} = 39057.5547154358
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
The graph
Graphing y = xarcctg(x)
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