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Plotting a function graph/ sin(x)*log(x)

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

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = sin(x)*log(x)
f(x)=log⁡(x)sin⁡(x)f{\left(x \right)} = \log{\left(x \right)} \sin{\left(x \right)} 
f = log(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)sin⁡(x)=0\log{\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=πx_{2} = \pi
Numerical solution
x1=−100.530964914873x_{1} = -100.530964914873
x2=34.5575191894877x_{2} = 34.5575191894877
x3=−31.4159265358979x_{3} = -31.4159265358979
x4=62.8318530717959x_{4} = 62.8318530717959
x5=−75.398223686155x_{5} = -75.398223686155
x6=−12.5663706143592x_{6} = -12.5663706143592
x7=−3.14159265358979x_{7} = -3.14159265358979
x8=−50.2654824574367x_{8} = -50.2654824574367
x9=−65.9734457253857x_{9} = -65.9734457253857
x10=−43.9822971502571x_{10} = -43.9822971502571
x11=28.2743338823081x_{11} = 28.2743338823081
x12=59.6902604182061x_{12} = 59.6902604182061
x13=56.5486677646163x_{13} = 56.5486677646163
x14=100.530964914873x_{14} = 100.530964914873
x15=−15.707963267949x_{15} = -15.707963267949
x16=−84.8230016469244x_{16} = -84.8230016469244
x17=15.707963267949x_{17} = 15.707963267949
x18=−91.106186954104x_{18} = -91.106186954104
x19=−37.6991118430775x_{19} = -37.6991118430775
x20=25.1327412287183x_{20} = 25.1327412287183
x21=−56.5486677646163x_{21} = -56.5486677646163
x22=−78.5398163397448x_{22} = -78.5398163397448
x23=40.8407044966673x_{23} = 40.8407044966673
x24=−18.8495559215388x_{24} = -18.8495559215388
x25=53.4070751110265x_{25} = 53.4070751110265
x26=47.1238898038469x_{26} = 47.1238898038469
x27=−72.2566310325652x_{27} = -72.2566310325652
x28=65.9734457253857x_{28} = 65.9734457253857
x29=−34.5575191894877x_{29} = -34.5575191894877
x30=12.5663706143592x_{30} = 12.5663706143592
x31=−28.2743338823081x_{31} = -28.2743338823081
x32=21.9911485751286x_{32} = 21.9911485751286
x33=−47.1238898038469x_{33} = -47.1238898038469
x34=−69.1150383789755x_{34} = -69.1150383789755
x35=81.6814089933346x_{35} = 81.6814089933346
x36=78.5398163397448x_{36} = 78.5398163397448
x37=−40.8407044966673x_{37} = -40.8407044966673
x38=138.230076757951x_{38} = 138.230076757951
x39=1x_{39} = 1
x40=−25.1327412287183x_{40} = -25.1327412287183
x41=97.3893722612836x_{41} = 97.3893722612836
x42=−97.3893722612836x_{42} = -97.3893722612836
x43=37.6991118430775x_{43} = 37.6991118430775
x44=−21.9911485751286x_{44} = -21.9911485751286
x45=3.14159265358979x_{45} = 3.14159265358979
x46=69.1150383789755x_{46} = 69.1150383789755
x47=−94.2477796076938x_{47} = -94.2477796076938
x48=−53.4070751110265x_{48} = -53.4070751110265
x49=−62.8318530717959x_{49} = -62.8318530717959
x50=−87.9645943005142x_{50} = -87.9645943005142
x51=−113.097335529233x_{51} = -113.097335529233
x52=−6.28318530717959x_{52} = -6.28318530717959
x53=84.8230016469244x_{53} = 84.8230016469244
x54=−81.6814089933346x_{54} = -81.6814089933346
x55=−59.6902604182061x_{55} = -59.6902604182061
x56=6.28318530717959x_{56} = 6.28318530717959
x57=87.9645943005142x_{57} = 87.9645943005142
x58=43.9822971502571x_{58} = 43.9822971502571
x59=−223.053078404875x_{59} = -223.053078404875
x60=−9.42477796076938x_{60} = -9.42477796076938
x61=31.4159265358979x_{61} = 31.4159265358979
x62=72.2566310325652x_{62} = 72.2566310325652
x63=94.2477796076938x_{63} = 94.2477796076938
x64=91.106186954104x_{64} = 91.106186954104
x65=9.42477796076938x_{65} = 9.42477796076938
x66=18.8495559215388x_{66} = 18.8495559215388
x67=50.2654824574367x_{67} = 50.2654824574367
x68=75.398223686155x_{68} = 75.398223686155
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(x)*log(x).
log⁡(0)sin⁡(0)\log{\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
log⁡(x)cos⁡(x)+sin⁡(x)x=0\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} = 0
Solve this equation
The roots of this equation
x1=32.9953908591221x_{1} = 32.9953908591221
x2=7.91497769383021x_{2} = 7.91497769383021
x3=26.7149311915258x_{3} = 26.7149311915258
x4=51.8411644567759x_{4} = 51.8411644567759
x5=86.3963937735675x_{5} = 86.3963937735675
x6=2.12761582523344x_{6} = 2.12761582523344
x7=45.5588408894342x_{7} = 45.5588408894342
x8=54.9824103570705x_{8} = 54.9824103570705
x9=89.5378754494563x_{9} = 89.5378754494563
x10=80.1134602593311x_{10} = 80.1134602593311
x11=17.2990352355066x_{11} = 17.2990352355066
x12=70.6891567862013x_{12} = 70.6891567862013
x13=95.8208633135828x_{13} = 95.8208633135828
x14=48.6999705880551x_{14} = 48.6999705880551
x15=39.2768442680313x_{15} = 39.2768442680313
x16=98.9623678062405x_{16} = 98.9623678062405
x17=64.406377021222x_{17} = 64.406377021222
x18=83.2549216304705x_{18} = 83.2549216304705
x19=23.5753663871051x_{19} = 23.5753663871051
x20=29.8549920106507x_{20} = 29.8549920106507
x21=11.0333063655933x_{21} = 11.0333063655933
x22=67.5477561419489x_{22} = 67.5477561419489
x23=73.8305759400225x_{23} = 73.8305759400225
x24=76.9720111193216x_{24} = 76.9720111193216
x25=14.1637961865355x_{25} = 14.1637961865355
x26=4.84255834039212x_{26} = 4.84255834039212
x27=61.2650231149052x_{27} = 61.2650231149052
x28=58.1236989891669x_{28} = 58.1236989891669
x29=20.4365678012128x_{29} = 20.4365678012128
x30=36.1360296011875x_{30} = 36.1360296011875
x31=42.4177914906586x_{31} = 42.4177914906586
x32=92.6793655993772x_{32} = 92.6793655993772
The values of the extrema at the points:
(32.99539085912214, 3.49623653326273)

(7.914977693830208, 2.06490964318559)

(26.7149311915258, 3.28500939657186)

(51.84116445677586, 3.94813739322056)

(86.3963937735675, -4.45893091363236)

(2.127615825233441, 0.640951613895412)

(45.55884088943418, 3.81894162090863)

(54.98241035707053, -4.00697204664365)

(89.5378754494563, 4.49464784936066)

(80.11346025933112, -4.38342611095494)

(17.2990352355066, -2.85006479973796)

(70.6891567862013, 4.2582686940799)

(95.82086331358285, 4.56246850547861)

(48.69997058805509, -3.88562417153593)

(39.27684426803133, 3.67054684507133)

(98.96236780624047, -4.59472854333644)

(64.40637702122196, 4.16518371214019)

(83.25492163047046, 4.42189093263579)

(23.57536638710508, -3.15991774048714)

(29.854992010650733, -3.39618690740209)

(11.03330636559327, -2.39920964673997)

(67.54775614194894, -4.21280883436135)

(73.83057594002254, -4.30175163100997)

(76.9720111193216, 4.3434224340588)

(14.16379618653552, 2.6497493761583)

(4.8425583403921175, -1.56409787578554)

(61.26502311490521, -4.11517672431722)

(58.12369898916687, 4.06253705090375)

(20.43656780121277, 3.01692915004008)

(36.13602960118748, -3.58718368340644)

(42.417791490658566, -3.74749373479586)

(92.67936559937723, -4.52913300203105)


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=86.3963937735675x_{1} = 86.3963937735675
x2=54.9824103570705x_{2} = 54.9824103570705
x3=80.1134602593311x_{3} = 80.1134602593311
x4=17.2990352355066x_{4} = 17.2990352355066
x5=48.6999705880551x_{5} = 48.6999705880551
x6=98.9623678062405x_{6} = 98.9623678062405
x7=23.5753663871051x_{7} = 23.5753663871051
x8=29.8549920106507x_{8} = 29.8549920106507
x9=11.0333063655933x_{9} = 11.0333063655933
x10=67.5477561419489x_{10} = 67.5477561419489
x11=73.8305759400225x_{11} = 73.8305759400225
x12=4.84255834039212x_{12} = 4.84255834039212
x13=61.2650231149052x_{13} = 61.2650231149052
x14=36.1360296011875x_{14} = 36.1360296011875
x15=42.4177914906586x_{15} = 42.4177914906586
x16=92.6793655993772x_{16} = 92.6793655993772
Maxima of the function at points:
x16=32.9953908591221x_{16} = 32.9953908591221
x16=7.91497769383021x_{16} = 7.91497769383021
x16=26.7149311915258x_{16} = 26.7149311915258
x16=51.8411644567759x_{16} = 51.8411644567759
x16=2.12761582523344x_{16} = 2.12761582523344
x16=45.5588408894342x_{16} = 45.5588408894342
x16=89.5378754494563x_{16} = 89.5378754494563
x16=70.6891567862013x_{16} = 70.6891567862013
x16=95.8208633135828x_{16} = 95.8208633135828
x16=39.2768442680313x_{16} = 39.2768442680313
x16=64.406377021222x_{16} = 64.406377021222
x16=83.2549216304705x_{16} = 83.2549216304705
x16=76.9720111193216x_{16} = 76.9720111193216
x16=14.1637961865355x_{16} = 14.1637961865355
x16=58.1236989891669x_{16} = 58.1236989891669
x16=20.4365678012128x_{16} = 20.4365678012128
Decreasing at intervals
[98.9623678062405,∞)\left[98.9623678062405, \infty\right)
Increasing at intervals
(−∞,4.84255834039212]\left(-\infty, 4.84255834039212\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
−log⁡(x)sin⁡(x)+2cos⁡(x)x−sin⁡(x)x2=0- \log{\left(x \right)} \sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=18.8855464491534x_{1} = 18.8855464491534
x2=100.535279615268x_{2} = 100.535279615268
x3=72.2630966850528x_{3} = 72.2630966850528
x4=31.4343721697806x_{4} = 31.4343721697806
x5=56.5574301916107x_{5} = 56.5574301916107
x6=3.53961476088587x_{6} = 3.53961476088587
x7=50.2756356438169x_{7} = 50.2756356438169
x8=15.7539096110127x_{8} = 15.7539096110127
x9=43.9943085957168x_{9} = 43.9943085957168
x10=59.6984521889897x_{10} = 59.6984521889897
x11=34.5738406188022x_{11} = 34.5738406188022
x12=81.686969567961x_{12} = 81.686969567961
x13=12.6285861720285x_{13} = 12.6285861720285
x14=22.0204948431363x_{14} = 22.0204948431363
x15=47.1349005959502x_{15} = 47.1349005959502
x16=37.7137169986599x_{16} = 37.7137169986599
x17=62.8395390693532x_{17} = 62.8395390693532
x18=87.9696723207031x_{18} = 87.9696723207031
x19=40.8538969938589x_{19} = 40.8538969938589
x20=6.4461035560751x_{20} = 6.4461035560751
x21=84.8283108211935x_{21} = 84.8283108211935
x22=75.4043590276847x_{22} = 75.4043590276847
x23=53.4164858945863x_{23} = 53.4164858945863
x24=69.1218687386001x_{24} = 69.1218687386001
x25=91.1110517789567x_{25} = 91.1110517789567
x26=65.9806806486246x_{26} = 65.9806806486246
x27=97.3938570020224x_{27} = 97.3938570020224
x28=25.1573740446396x_{28} = 25.1573740446396
x29=78.5456512461642x_{29} = 78.5456512461642
x30=9.51732588699837x_{30} = 9.51732588699837
x31=94.2524472357136x_{31} = 94.2524472357136
x32=28.2954682170335x_{32} = 28.2954682170335

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
[97.3938570020224,∞)\left[97.3938570020224, \infty\right)
Convex at the intervals
(−∞,3.53961476088587]\left(-\infty, 3.53961476088587\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(log⁡(x)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\log{\left(x \right)} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(log⁡(x)sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\log{\left(x \right)} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)*log(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(x)sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(x)sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{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:
log⁡(x)sin⁡(x)=−log⁡(−x)sin⁡(x)\log{\left(x \right)} \sin{\left(x \right)} = - \log{\left(- x \right)} \sin{\left(x \right)}
- No
log⁡(x)sin⁡(x)=log⁡(−x)sin⁡(x)\log{\left(x \right)} \sin{\left(x \right)} = \log{\left(- x \right)} \sin{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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