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Plotting a function graph/ ctg3x/5

Graphing y = ctg3x/5

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       cot(3*x)
f(x) = --------
          5
f(x)=cot⁡(3x)5f{\left(x \right)} = \frac{\cot{\left(3 x \right)}}{5} 
f = cot(3*x)/5
The graph of the function
02468-8-6-4-2-1010-100100
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:
cot⁡(3x)5=0\frac{\cot{\left(3 x \right)}}{5} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π6x_{1} = \frac{\pi}{6}
Numerical solution
x1=−12.0427718387609x_{1} = -12.0427718387609
x2=−23.5619449019235x_{2} = -23.5619449019235
x3=44.5058959258554x_{3} = 44.5058959258554
x4=−29.845130209103x_{4} = -29.845130209103
x5=−21.4675497995303x_{5} = -21.4675497995303
x6=42.4115008234622x_{6} = 42.4115008234622
x7=46.6002910282486x_{7} = 46.6002910282486
x8=36.1283155162826x_{8} = 36.1283155162826
x9=−34.0339204138894x_{9} = -34.0339204138894
x10=34.0339204138894x_{10} = 34.0339204138894
x11=2.61799387799149x_{11} = 2.61799387799149
x12=−5.75958653158129x_{12} = -5.75958653158129
x13=66.497044500984x_{13} = 66.497044500984
x14=−41.3643032722656x_{14} = -41.3643032722656
x15=5.75958653158129x_{15} = 5.75958653158129
x16=9.94837673636768x_{16} = 9.94837673636768
x17=−80.1106126665397x_{17} = -80.1106126665397
x18=86.3937979737193x_{18} = 86.3937979737193
x19=64.4026493985908x_{19} = 64.4026493985908
x20=−95.8185759344887x_{20} = -95.8185759344887
x21=47.6474885794452x_{21} = 47.6474885794452
x22=90.5825881785057x_{22} = 90.5825881785057
x23=60.2138591938044x_{23} = 60.2138591938044
x24=−1.5707963267949x_{24} = -1.5707963267949
x25=−18.3259571459405x_{25} = -18.3259571459405
x26=71.733032256967x_{26} = 71.733032256967
x27=−60.2138591938044x_{27} = -60.2138591938044
x28=−78.0162175641465x_{28} = -78.0162175641465
x29=53.9306738866248x_{29} = 53.9306738866248
x30=−19.3731546971371x_{30} = -19.3731546971371
x31=73.8274273593601x_{31} = 73.8274273593601
x32=31.9395253114962x_{32} = 31.9395253114962
x33=−40.317105721069x_{33} = -40.317105721069
x34=12.0427718387609x_{34} = 12.0427718387609
x35=84.2994028713261x_{35} = 84.2994028713261
x36=78.0162175641465x_{36} = 78.0162175641465
x37=−93.7241808320955x_{37} = -93.7241808320955
x38=22.5147473507269x_{38} = 22.5147473507269
x39=−3.66519142918809x_{39} = -3.66519142918809
x40=−27.7507351067098x_{40} = -27.7507351067098
x41=−65.4498469497874x_{41} = -65.4498469497874
x42=−38.2227106186758x_{42} = -38.2227106186758
x43=100.007366139275x_{43} = 100.007366139275
x44=97.9129710368819x_{44} = 97.9129710368819
x45=49.7418836818384x_{45} = 49.7418836818384
x46=−56.025068989018x_{46} = -56.025068989018
x47=−62.3082542961976x_{47} = -62.3082542961976
x48=−51.8362787842316x_{48} = -51.8362787842316
x49=58.1194640914112x_{49} = 58.1194640914112
x50=−87.4409955249159x_{50} = -87.4409955249159
x51=−73.8274273593601x_{51} = -73.8274273593601
x52=51.8362787842316x_{52} = 51.8362787842316
x53=−85.3466004225227x_{53} = -85.3466004225227
x54=27.7507351067098x_{54} = 27.7507351067098
x55=−45.553093477052x_{55} = -45.553093477052
x56=29.845130209103x_{56} = 29.845130209103
x57=75.9218224617533x_{57} = 75.9218224617533
x58=80.1106126665397x_{58} = 80.1106126665397
x59=−58.1194640914112x_{59} = -58.1194640914112
x60=−49.7418836818384x_{60} = -49.7418836818384
x61=−36.1283155162826x_{61} = -36.1283155162826
x62=−71.733032256967x_{62} = -71.733032256967
x63=16.2315620435473x_{63} = 16.2315620435473
x64=24.60914245312x_{64} = 24.60914245312
x65=−100.007366139275x_{65} = -100.007366139275
x66=−31.9395253114962x_{66} = -31.9395253114962
x67=−9.94837673636768x_{67} = -9.94837673636768
x68=−97.9129710368819x_{68} = -97.9129710368819
x69=38.2227106186758x_{69} = 38.2227106186758
x70=69.6386371545737x_{70} = 69.6386371545737
x71=3.66519142918809x_{71} = 3.66519142918809
x72=−67.5442420521806x_{72} = -67.5442420521806
x73=−43.4586983746588x_{73} = -43.4586983746588
x74=62.3082542961976x_{74} = 62.3082542961976
x75=−7.85398163397448x_{75} = -7.85398163397448
x76=82.2050077689329x_{76} = 82.2050077689329
x77=7.85398163397448x_{77} = 7.85398163397448
x78=−14.1371669411541x_{78} = -14.1371669411541
x79=25.6563400043166x_{79} = 25.6563400043166
x80=18.3259571459405x_{80} = 18.3259571459405
x81=−53.9306738866248x_{81} = -53.9306738866248
x82=−75.9218224617533x_{82} = -75.9218224617533
x83=93.7241808320955x_{83} = 93.7241808320955
x84=−91.6297857297023x_{84} = -91.6297857297023
x85=14.1371669411541x_{85} = 14.1371669411541
x86=−16.2315620435473x_{86} = -16.2315620435473
x87=−82.2050077689329x_{87} = -82.2050077689329
x88=40.317105721069x_{88} = 40.317105721069
x89=20.4203522483337x_{89} = 20.4203522483337
x90=−69.6386371545737x_{90} = -69.6386371545737
x91=95.8185759344887x_{91} = 95.8185759344887
x92=−89.5353906273091x_{92} = -89.5353906273091
x93=−47.6474885794452x_{93} = -47.6474885794452
x94=92.6769832808989x_{94} = 92.6769832808989
x95=68.5914396033772x_{95} = 68.5914396033772
x96=88.4881930761125x_{96} = 88.4881930761125
x97=−25.6563400043166x_{97} = -25.6563400043166
x98=−63.3554518473942x_{98} = -63.3554518473942
x99=56.025068989018x_{99} = 56.025068989018
x100=−84.2994028713261x_{100} = -84.2994028713261
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cot(3*x)/5.
cot⁡(0⋅3)5\frac{\cot{\left(0 \cdot 3 \right)}}{5}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
−3cot⁡2(3x)5−35=0- \frac{3 \cot^{2}{\left(3 x \right)}}{5} - \frac{3}{5} = 0
Solve this equation
Solutions are not found,
function may have no extrema
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
18(cot⁡2(3x)+1)cot⁡(3x)5=0\frac{18 \left(\cot^{2}{\left(3 x \right)} + 1\right) \cot{\left(3 x \right)}}{5} = 0
Solve this equation
The roots of this equation
x1=π6x_{1} = \frac{\pi}{6}

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
(−∞,π6]\left(-\infty, \frac{\pi}{6}\right]
Convex at the intervals
[π6,∞)\left[\frac{\pi}{6}, \infty\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→−∞(cot⁡(3x)5)y = \lim_{x \to -\infty}\left(\frac{\cot{\left(3 x \right)}}{5}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(cot⁡(3x)5)y = \lim_{x \to \infty}\left(\frac{\cot{\left(3 x \right)}}{5}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(3*x)/5, divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(cot⁡(3x)5x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(3 x \right)}}{5 x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(3x)5x)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(3 x \right)}}{5 x}\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:
cot⁡(3x)5=−cot⁡(3x)5\frac{\cot{\left(3 x \right)}}{5} = - \frac{\cot{\left(3 x \right)}}{5}
- No
cot⁡(3x)5=cot⁡(3x)5\frac{\cot{\left(3 x \right)}}{5} = \frac{\cot{\left(3 x \right)}}{5}
- No
so, the function
not is
neither even, nor odd
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