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  • Graphing y =:
  • x^3-((x^4)/4)
  • -x^3+3x+2
  • x^3-3x^2+6
  • (x^2-1)/(x-1)

  • Identical expressions

  • (cot(x)*log(x))*sin(x)
  • ( cotangent of (x) multiply by logarithm of (x)) multiply by sinus of (x)
  • (cot(x)log(x))sin(x)
  • cotxlogxsinx

  • Similar expressions

  • (cot(x)*log(x))*sinx
Plotting a function graph/ (cot(x)*log(x))*sin(x)

Graphing y = (cot(x)*log(x))*sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = cot(x)*log(x)*sin(x)
f(x)=log⁡(x)cot⁡(x)sin⁡(x)f{\left(x \right)} = \log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)} 
f = (log(x)*cot(x))*sin(x)
The graph of the function
02468-8-6-4-2-10105-5
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:
log⁡(x)cot⁡(x)sin⁡(x)=0\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Numerical solution
x1=29.845130209103x_{1} = 29.845130209103
x2=14.1371669411541x_{2} = 14.1371669411541
x3=48.6946861306418x_{3} = 48.6946861306418
x4=−36.1283155162826x_{4} = -36.1283155162826
x5=10.9955742875643x_{5} = 10.9955742875643
x6=−58.1194640914111x_{6} = -58.1194640914111
x7=−70.6858347057703x_{7} = -70.6858347057703
x8=67.5442420521806x_{8} = 67.5442420521806
x9=95.8185759344887x_{9} = 95.8185759344887
x10=−48.6946861306418x_{10} = -48.6946861306418
x11=80.1106126665397x_{11} = 80.1106126665397
x12=32.9867228626928x_{12} = 32.9867228626928
x13=−20.4203522483337x_{13} = -20.4203522483337
x14=7.85398163397448x_{14} = 7.85398163397448
x15=−51.8362787842316x_{15} = -51.8362787842316
x16=70.6858347057703x_{16} = 70.6858347057703
x17=−29.845130209103x_{17} = -29.845130209103
x18=−4.71238898038469x_{18} = -4.71238898038469
x19=−32.9867228626928x_{19} = -32.9867228626928
x20=−39.2699081698724x_{20} = -39.2699081698724
x21=−86.3937979737193x_{21} = -86.3937979737193
x22=17.2787595947439x_{22} = 17.2787595947439
x23=39.2699081698724x_{23} = 39.2699081698724
x24=−32.9867228626928x_{24} = -32.9867228626928
x25=42.4115008234622x_{25} = 42.4115008234622
x26=−64.4026493985908x_{26} = -64.4026493985908
x27=−42.4115008234622x_{27} = -42.4115008234622
x28=−17.2787595947439x_{28} = -17.2787595947439
x29=−1.5707963267949x_{29} = -1.5707963267949
x30=98.9601685880785x_{30} = 98.9601685880785
x31=45.553093477052x_{31} = 45.553093477052
x32=−42.4115008234622x_{32} = -42.4115008234622
x33=−14.1371669411541x_{33} = -14.1371669411541
x34=58.1194640914112x_{34} = 58.1194640914112
x35=−10.9955742875643x_{35} = -10.9955742875643
x36=−26.7035375555132x_{36} = -26.7035375555132
x37=−95.8185759344887x_{37} = -95.8185759344887
x38=−89.5353906273091x_{38} = -89.5353906273091
x39=4.71238898038469x_{39} = 4.71238898038469
x40=64.4026493985908x_{40} = 64.4026493985908
x41=61.261056745001x_{41} = 61.261056745001
x42=54.9778714378214x_{42} = 54.9778714378214
x43=−83.2522053201295x_{43} = -83.2522053201295
x44=83.2522053201295x_{44} = 83.2522053201295
x45=1.5707963267949x_{45} = 1.5707963267949
x46=−67.5442420521806x_{46} = -67.5442420521806
x47=−76.9690200129499x_{47} = -76.9690200129499
x48=−80.1106126665397x_{48} = -80.1106126665397
x49=−45.553093477052x_{49} = -45.553093477052
x50=20.4203522483337x_{50} = 20.4203522483337
x51=23.5619449019235x_{51} = 23.5619449019235
x52=−98.9601685880785x_{52} = -98.9601685880785
x53=92.6769832808989x_{53} = 92.6769832808989
x54=76.9690200129499x_{54} = 76.9690200129499
x55=86.3937979737193x_{55} = 86.3937979737193
x56=−92.6769832808989x_{56} = -92.6769832808989
x57=89.5353906273091x_{57} = 89.5353906273091
x58=−73.8274273593601x_{58} = -73.8274273593601
x59=−7.85398163397448x_{59} = -7.85398163397448
x60=26.7035375555132x_{60} = 26.7035375555132
x61=−54.9778714378214x_{61} = -54.9778714378214
x62=36.1283155162826x_{62} = 36.1283155162826
x63=51.8362787842316x_{63} = 51.8362787842316
x64=73.8274273593601x_{64} = 73.8274273593601
x65=−39.2699081698724x_{65} = -39.2699081698724
x66=−23.5619449019235x_{66} = -23.5619449019235
x67=−61.261056745001x_{67} = -61.261056745001
x68=−4.71238898038469x_{68} = -4.71238898038469
x69=−61.261056745001x_{69} = -61.261056745001
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (cot(x)*log(x))*sin(x).
log⁡(0)cot⁡(0)sin⁡(0)\log{\left(0 \right)} \cot{\left(0 \right)} \sin{\left(0 \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
((−cot⁡2(x)−1)log⁡(x)+cot⁡(x)x)sin⁡(x)+log⁡(x)cos⁡(x)cot⁡(x)=0\left(\left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(x \right)} + \frac{\cot{\left(x \right)}}{x}\right) \sin{\left(x \right)} + \log{\left(x \right)} \cos{\left(x \right)} \cot{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=34.5656848442796x_{1} = 34.5656848442796
x2=6.36781151369107x_{2} = 6.36781151369107
x3=18.8675971617309x_{3} = 18.8675971617309
x4=62.8356966541501x_{4} = 62.8356966541501
x5=94.2501135627054x_{5} = 94.2501135627054
x6=37.7064180281721x_{6} = 37.7064180281721
x7=100.533122377741x_{7} = 100.533122377741
x8=28.2849113047725x_{8} = 28.2849113047725
x9=40.8473034495909x_{9} = 40.8473034495909
x10=59.6943570030875x_{10} = 59.6943570030875
x11=75.4012916642681x_{11} = 75.4012916642681
x12=25.1450734377105x_{12} = 25.1450734377105
x13=56.5530498251275x_{13} = 56.5530498251275
x14=43.9883049460921x_{14} = 43.9883049460921
x15=50.27056033759x_{15} = 50.27056033759
x16=65.9770636783598x_{16} = 65.9770636783598
x17=12.5976921976804x_{17} = 12.5976921976804
x18=69.1184539759405x_{18} = 69.1184539759405
x19=87.967133489911x_{19} = 87.967133489911
x20=15.7310277752208x_{20} = 15.7310277752208
x21=47.1293968198114x_{21} = 47.1293968198114
x22=31.4251563350128x_{22} = 31.4251563350128
x23=53.4117815402062x_{23} = 53.4117815402062
x24=22.0058475927713x_{24} = 22.0058475927713
x25=97.3916147574604x_{25} = 97.3916147574604
x26=78.5427340593526x_{26} = 78.5427340593526
x27=3.37991614208723x_{27} = 3.37991614208723
x28=1.27285069827148x_{28} = 1.27285069827148
x29=9.47170218677955x_{29} = 9.47170218677955
x30=84.8256564376189x_{30} = 84.8256564376189
x31=72.2598642156451x_{31} = 72.2598642156451
x32=91.1086195251935x_{32} = 91.1086195251935
x33=81.6841895128946x_{33} = 81.6841895128946
The values of the extrema at the points:
(34.56568484427963, -3.54274330777479)

(6.367811513691074, 1.8446308321891)

(18.86759716173087, 2.93696797853021)

(62.835696654150134, 4.14049274584816)

(94.25011356270541, 4.54593964955556)

(37.70641802817207, 3.62973343908044)

(100.53312237774094, 4.61047651900993)

(28.284911304772503, -3.34214152179011)

(40.847303449590925, -3.70976003369716)

(59.69435700308747, -4.08920318061012)

(75.40129166426813, 4.32280406143887)

(25.14507343771052, 3.22441678455125)

(56.553049825127495, 4.03514038997721)

(43.98830494609213, 3.78385551437724)

(50.27056033759003, 3.91736911923955)

(65.9770636783598, -4.18927974348899)

(12.597692197680386, 2.53227099874907)

(69.11845397594048, 4.23579704883419)

(87.96713348991098, 4.4769488290828)

(15.731027775220827, -2.7549021263166)

(47.1293968198114, -3.85283851894378)

(31.425156335012787, 3.44746188086714)

(53.41178154020617, -3.9779872921632)

(22.00584759277127, -3.09097426796676)

(97.39161475746043, -4.57872860340226)

(78.5427340593526, -4.3636242855634)

(3.3799161420872266, -1.18342849059061)

(1.2728506982714773, 0.0708232692475832)

(9.471702186779549, -2.24583383410247)

(84.82565643761893, -4.44058240090197)

(72.25986421564514, -4.28024647479203)

(91.10861952519349, -4.5120390660658)

(81.68418951289463, 4.40284344444233)


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=34.5656848442796x_{1} = 34.5656848442796
x2=28.2849113047725x_{2} = 28.2849113047725
x3=40.8473034495909x_{3} = 40.8473034495909
x4=59.6943570030875x_{4} = 59.6943570030875
x5=65.9770636783598x_{5} = 65.9770636783598
x6=15.7310277752208x_{6} = 15.7310277752208
x7=47.1293968198114x_{7} = 47.1293968198114
x8=53.4117815402062x_{8} = 53.4117815402062
x9=22.0058475927713x_{9} = 22.0058475927713
x10=97.3916147574604x_{10} = 97.3916147574604
x11=78.5427340593526x_{11} = 78.5427340593526
x12=3.37991614208723x_{12} = 3.37991614208723
x13=9.47170218677955x_{13} = 9.47170218677955
x14=84.8256564376189x_{14} = 84.8256564376189
x15=72.2598642156451x_{15} = 72.2598642156451
x16=91.1086195251935x_{16} = 91.1086195251935
Maxima of the function at points:
x16=6.36781151369107x_{16} = 6.36781151369107
x16=18.8675971617309x_{16} = 18.8675971617309
x16=62.8356966541501x_{16} = 62.8356966541501
x16=94.2501135627054x_{16} = 94.2501135627054
x16=37.7064180281721x_{16} = 37.7064180281721
x16=100.533122377741x_{16} = 100.533122377741
x16=75.4012916642681x_{16} = 75.4012916642681
x16=25.1450734377105x_{16} = 25.1450734377105
x16=56.5530498251275x_{16} = 56.5530498251275
x16=43.9883049460921x_{16} = 43.9883049460921
x16=50.27056033759x_{16} = 50.27056033759
x16=12.5976921976804x_{16} = 12.5976921976804
x16=69.1184539759405x_{16} = 69.1184539759405
x16=87.967133489911x_{16} = 87.967133489911
x16=31.4251563350128x_{16} = 31.4251563350128
x16=1.27285069827148x_{16} = 1.27285069827148
x16=81.6841895128946x_{16} = 81.6841895128946
Decreasing at intervals
[97.3916147574604,∞)\left[97.3916147574604, \infty\right)
Increasing at intervals
(−∞,3.37991614208723]\left(-\infty, 3.37991614208723\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((cot⁡2(x)+1)log⁡(x)−cot⁡(x)x)cos⁡(x)+(−2(cot⁡2(x)+1)log⁡(x)cot⁡(x)+2(cot⁡2(x)+1)x+cot⁡(x)x2)sin⁡(x)+log⁡(x)sin⁡(x)cot⁡(x))=0- (2 \left(\left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} - \frac{\cot{\left(x \right)}}{x}\right) \cos{\left(x \right)} + \left(- 2 \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \cot{\left(x \right)} + \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{x} + \frac{\cot{\left(x \right)}}{x^{2}}\right) \sin{\left(x \right)} + \log{\left(x \right)} \sin{\left(x \right)} \cot{\left(x \right)}) = 0
Solve this equation
The roots of this equation
x1=70.6924780956907x_{1} = 70.6924780956907
x2=36.1437352734116x_{2} = 36.1437352734116
x3=33.0040471601321x_{3} = 33.0040471601321
x4=48.7052521633042x_{4} = 48.7052521633042
x5=76.9750016538421x_{5} = 76.9750016538421
x6=42.4240774344727x_{6} = 42.4240774344727
x7=73.8337238588912x_{7} = 73.8337238588912
x8=14.1901528741925x_{8} = 14.1901528741925
x9=80.1163073549944x_{9} = 80.1163073549944
x10=2.28203436188726x_{10} = 2.28203436188726
x11=89.5403599344556x_{11} = 89.5403599344556
x12=86.3989891915502x_{12} = 86.3989891915502
x13=7.97332415905512x_{13} = 7.97332415905512
x14=23.5887477756706x_{14} = 23.5887477756706
x15=98.9645667863943x_{15} = 98.9645667863943
x16=39.2837741382964x_{16} = 39.2837741382964
x17=4.95364945549859x_{17} = 4.95364945549859
x18=54.9869474043708x_{18} = 54.9869474043708
x19=92.681747618866x_{19} = 92.681747618866
x20=95.8231504264293x_{20} = 95.8231504264293
x21=51.8460478223206x_{21} = 51.8460478223206
x22=29.8648369783603x_{22} = 29.8648369783603
x23=26.7262995386281x_{23} = 26.7262995386281
x24=61.2689882085312x_{24} = 61.2689882085312
x25=17.3191834025164x_{25} = 17.3191834025164
x26=11.0703232999089x_{26} = 11.0703232999089
x27=20.4527151636554x_{27} = 20.4527151636554
x28=83.2576375062293x_{28} = 83.2576375062293
x29=58.127932349539x_{29} = 58.127932349539
x30=64.4101035740511x_{30} = 64.4101035740511
x31=45.5645846448375x_{31} = 45.5645846448375
x32=67.5512693271385x_{32} = 67.5512693271385

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
[95.8231504264293,∞)\left[95.8231504264293, \infty\right)
Convex at the intervals
(−∞,2.28203436188726]\left(-\infty, 2.28203436188726\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→−∞(log⁡(x)cot⁡(x)sin⁡(x))y = \lim_{x \to -\infty}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(log⁡(x)cot⁡(x)sin⁡(x))y = \lim_{x \to \infty}\left(\log{\left(x \right)} \cot{\left(x \right)} \sin{\left(x \right)}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (cot(x)*log(x))*sin(x), 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→−∞(log⁡(x)sin⁡(x)cot⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} \sin{\left(x \right)} \cot{\left(x \right)}}{x}\right)
True

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