Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • (x+1)^3/(x-1)^2
  • 4x^2-x^2
  • 3x^2+2x-1
  • x+x
  • Derivative of:
  • log(x)^2*sin(x) log(x)^2*sin(x)

  • Identical expressions

  • log(x)^ two *sin(x)
  • logarithm of (x) squared multiply by sinus of (x)
  • logarithm of (x) to the power of two multiply by sinus of (x)
  • log(x)2*sin(x)
  • logx2*sinx
  • log(x)²*sin(x)
  • log(x) to the power of 2*sin(x)
  • log(x)^2sin(x)
  • log(x)2sin(x)
  • logx2sinx
  • logx^2sinx

  • Similar expressions

  • log(x)^2*sinx
Plotting a function graph/ log(x)^2*sin(x)

Graphing y = log(x)^2*sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          2          
f(x) = log (x)*sin(x)
f(x)=log⁡(x)2sin⁡(x)f{\left(x \right)} = \log{\left(x \right)}^{2} \sin{\left(x \right)} 
f = log(x)^2*sin(x)
The graph of the function
02468-8-6-4-2-1010-1010
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)2sin⁡(x)=0\log{\left(x \right)}^{2} \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=−75.398223686155x_{1} = -75.398223686155
x2=34.5575191894877x_{2} = 34.5575191894877
x3=62.8318530717959x_{3} = 62.8318530717959
x4=−91.106186954104x_{4} = -91.106186954104
x5=−43.9822971502571x_{5} = -43.9822971502571
x6=28.2743338823081x_{6} = 28.2743338823081
x7=−9.42477796076938x_{7} = -9.42477796076938
x8=−12.5663706143592x_{8} = -12.5663706143592
x9=59.6902604182061x_{9} = 59.6902604182061
x10=56.5486677646163x_{10} = 56.5486677646163
x11=100.530964914873x_{11} = 100.530964914873
x12=15.707963267949x_{12} = 15.707963267949
x13=−53.4070751110265x_{13} = -53.4070751110265
x14=−6.28318530717959x_{14} = -6.28318530717959
x15=25.1327412287183x_{15} = 25.1327412287183
x16=−47.1238898038469x_{16} = -47.1238898038469
x17=40.8407044966673x_{17} = 40.8407044966673
x18=53.4070751110265x_{18} = 53.4070751110265
x19=47.1238898038469x_{19} = 47.1238898038469
x20=−97.3893722612836x_{20} = -97.3893722612836
x21=−72.2566310325652x_{21} = -72.2566310325652
x22=65.9734457253857x_{22} = 65.9734457253857
x23=−81.6814089933346x_{23} = -81.6814089933346
x24=12.5663706143592x_{24} = 12.5663706143592
x25=21.9911485751286x_{25} = 21.9911485751286
x26=−116.238928182822x_{26} = -116.238928182822
x27=−3.14159265358979x_{27} = -3.14159265358979
x28=−62.8318530717959x_{28} = -62.8318530717959
x29=−40.8407044966673x_{29} = -40.8407044966673
x30=81.6814089933346x_{30} = 81.6814089933346
x31=78.5398163397448x_{31} = 78.5398163397448
x32=−87.9645943005142x_{32} = -87.9645943005142
x33=−84.8230016469244x_{33} = -84.8230016469244
x34=−18.8495559215388x_{34} = -18.8495559215388
x35=−69.1150383789755x_{35} = -69.1150383789755
x36=97.3893722612836x_{36} = 97.3893722612836
x37=37.6991118430775x_{37} = 37.6991118430775
x38=3.14159265358979x_{38} = 3.14159265358979
x39=69.1150383789755x_{39} = 69.1150383789755
x40=−34.5575191894877x_{40} = -34.5575191894877
x41=−56.5486677646163x_{41} = -56.5486677646163
x42=−94.2477796076938x_{42} = -94.2477796076938
x43=−78.5398163397448x_{43} = -78.5398163397448
x44=84.8230016469244x_{44} = 84.8230016469244
x45=−21.9911485751286x_{45} = -21.9911485751286
x46=−25.1327412287183x_{46} = -25.1327412287183
x47=−65.9734457253857x_{47} = -65.9734457253857
x48=−100.530964914873x_{48} = -100.530964914873
x49=6.28318530717959x_{49} = 6.28318530717959
x50=−28.2743338823081x_{50} = -28.2743338823081
x51=87.9645943005142x_{51} = 87.9645943005142
x52=43.9822971502571x_{52} = 43.9822971502571
x53=−59.6902604182061x_{53} = -59.6902604182061
x54=−50.2654824574367x_{54} = -50.2654824574367
x55=31.4159265358979x_{55} = 31.4159265358979
x56=72.2566310325652x_{56} = 72.2566310325652
x57=−37.6991118430775x_{57} = -37.6991118430775
x58=94.2477796076938x_{58} = 94.2477796076938
x59=91.106186954104x_{59} = 91.106186954104
x60=9.42477796076938x_{60} = 9.42477796076938
x61=−15.707963267949x_{61} = -15.707963267949
x62=−31.4159265358979x_{62} = -31.4159265358979
x63=18.8495559215388x_{63} = 18.8495559215388
x64=50.2654824574367x_{64} = 50.2654824574367
x65=75.398223686155x_{65} = 75.398223686155
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(x)^2*sin(x).
log⁡(0)2sin⁡(0)\log{\left(0 \right)}^{2} \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)2cos⁡(x)+2log⁡(x)sin⁡(x)x=0\log{\left(x \right)}^{2} \cos{\left(x \right)} + \frac{2 \log{\left(x \right)} \sin{\left(x \right)}}{x} = 0
Solve this equation
The roots of this equation
x1=64.4101040053417x_{1} = 64.4101040053417
x2=70.6924784078238x_{2} = 70.6924784078238
x3=33.0040517046886x_{3} = 33.0040517046886
x4=98.9645668841232x_{4} = 98.9645668841232
x5=7.97420032258306x_{5} = 7.97420032258306
x6=42.424079298118x_{6} = 42.424079298118
x7=36.1437385612721x_{7} = 36.1437385612721
x8=23.5887629853956x_{8} = 23.5887629853956
x9=29.8648434760199x_{9} = 29.8648434760199
x10=95.8231505356154x_{10} = 95.8231505356154
x11=61.2689887218475x_{11} = 61.2689887218475
x12=86.3989893474972x_{12} = 86.3989893474972
x13=17.3192304590556x_{13} = 17.3192304590556
x14=4.95912418553785x_{14} = 4.95912418553785
x15=26.7263092231694x_{15} = 26.7263092231694
x16=11.0705736478894x_{16} = 11.0705736478894
x17=58.1279329662983x_{17} = 58.1279329662983
x18=48.7052533091617x_{18} = 48.7052533091617
x19=67.5512696926238x_{19} = 67.5512696926238
x20=2.35289421180674x_{20} = 2.35289421180674
x21=83.257637683432x_{21} = 83.257637683432
x22=76.9750018862394x_{22} = 76.9750018862394
x23=14.190251384224x_{23} = 14.190251384224
x24=89.5403600723421x_{24} = 89.5403600723421
x25=80.1163075573717x_{25} = 80.1163075573717
x26=92.6817477413168x_{26} = 92.6817477413168
x27=51.8460487425212x_{27} = 51.8460487425212
x28=73.8337241273519x_{28} = 73.8337241273519
x29=39.2837765846036x_{29} = 39.2837765846036
x30=54.9869481532823x_{30} = 54.9869481532823
x31=45.5645860935039x_{31} = 45.5645860935039
x32=20.4527407254987x_{32} = 20.4527407254987
The values of the extrema at the points:
(64.41010400534174, 17.3489964024935)

(70.69247840782383, 18.1330523774875)

(33.00405170468864, 12.2245880017778)

(98.96456688412323, -21.1116324913823)

(7.974200322583059, 4.27954122137984)

(42.42407929811798, -14.0442649325846)

(36.14373856127208, -12.8686523003517)

(23.588762985395583, -9.98687745696934)

(29.86484347601995, -11.5352067800173)

(95.82315053561537, 20.8162277723952)

(61.26898872184754, -16.9349458688333)

(86.39898934749725, -19.8821988564884)

(17.319230459055635, -8.12620342819105)

(4.9591241855378465, -2.48628597250277)

(26.726309223169388, 10.7926868279148)

(11.07057364788943, -5.76436304513365)

(58.12793296629834, 16.5045032554775)

(48.70525330916171, -15.0984967660752)

(67.55126969262382, -17.7479774247625)

(2.3528942118067393, 0.519400036707966)

(83.25763768343197, 19.5532636839555)

(76.97500188623944, 18.865487215289)

(14.190251384223993, 7.02613780609268)

(89.54036007234214, 20.2019840187013)

(80.11630755737171, -19.2145802690323)

(92.68174774131678, -20.5131621670215)

(51.846048742521226, 15.5881609091595)

(73.83372412735193, -18.5052505363528)

(39.283776584603615, 13.4735621709123)

(54.98694815328231, -16.0561557271752)

(45.56458609350386, 14.5847967872788)

(20.4527407254987, 9.1042522581078)


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=98.9645668841232x_{1} = 98.9645668841232
x2=42.424079298118x_{2} = 42.424079298118
x3=36.1437385612721x_{3} = 36.1437385612721
x4=23.5887629853956x_{4} = 23.5887629853956
x5=29.8648434760199x_{5} = 29.8648434760199
x6=61.2689887218475x_{6} = 61.2689887218475
x7=86.3989893474972x_{7} = 86.3989893474972
x8=17.3192304590556x_{8} = 17.3192304590556
x9=4.95912418553785x_{9} = 4.95912418553785
x10=11.0705736478894x_{10} = 11.0705736478894
x11=48.7052533091617x_{11} = 48.7052533091617
x12=67.5512696926238x_{12} = 67.5512696926238
x13=80.1163075573717x_{13} = 80.1163075573717
x14=92.6817477413168x_{14} = 92.6817477413168
x15=73.8337241273519x_{15} = 73.8337241273519
x16=54.9869481532823x_{16} = 54.9869481532823
Maxima of the function at points:
x16=64.4101040053417x_{16} = 64.4101040053417
x16=70.6924784078238x_{16} = 70.6924784078238
x16=33.0040517046886x_{16} = 33.0040517046886
x16=7.97420032258306x_{16} = 7.97420032258306
x16=95.8231505356154x_{16} = 95.8231505356154
x16=26.7263092231694x_{16} = 26.7263092231694
x16=58.1279329662983x_{16} = 58.1279329662983
x16=2.35289421180674x_{16} = 2.35289421180674
x16=83.257637683432x_{16} = 83.257637683432
x16=76.9750018862394x_{16} = 76.9750018862394
x16=14.190251384224x_{16} = 14.190251384224
x16=89.5403600723421x_{16} = 89.5403600723421
x16=51.8460487425212x_{16} = 51.8460487425212
x16=39.2837765846036x_{16} = 39.2837765846036
x16=45.5645860935039x_{16} = 45.5645860935039
x16=20.4527407254987x_{16} = 20.4527407254987
Decreasing at intervals
[98.9645668841232,∞)\left[98.9645668841232, \infty\right)
Increasing at intervals
(−∞,4.95912418553785]\left(-\infty, 4.95912418553785\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)2sin⁡(x)+4log⁡(x)cos⁡(x)x−2(log⁡(x)−1)sin⁡(x)x2=0- \log{\left(x \right)}^{2} \sin{\left(x \right)} + \frac{4 \log{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{2 \left(\log{\left(x \right)} - 1\right) \sin{\left(x \right)}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=72.2695600625994x_{1} = 72.2695600625994
x2=1.65706451750942x_{2} = 1.65706451750942
x3=3.79739269085901x_{3} = 3.79739269085901
x4=37.7282990086415x_{4} = 37.7282990086415
x5=53.4258900441622x_{5} = 53.4258900441622
x6=47.145901016022x_{6} = 47.145901016022
x7=31.4527729390447x_{7} = 31.4527729390447
x8=6.59122807282926x_{8} = 6.59122807282926
x9=59.7066394917018x_{9} = 59.7066394917018
x10=81.6925286637411x_{10} = 81.6925286637411
x11=97.3983409430341x_{11} = 97.3983409430341
x12=65.9879124364584x_{12} = 65.9879124364584
x13=25.181905221086x_{13} = 25.181905221086
x14=87.9747491999982x_{14} = 87.9747491999982
x15=84.8336186997604x_{15} = 84.8336186997604
x16=100.53959359966x_{16} = 100.53959359966
x17=62.8472213411914x_{17} = 62.8472213411914
x18=69.1286964375608x_{18} = 69.1286964375608
x19=22.0496746158449x_{19} = 22.0496746158449
x20=18.9212415222179x_{20} = 18.9212415222179
x21=40.867072151817x_{21} = 40.867072151817
x22=75.4104924106057x_{22} = 75.4104924106057
x23=44.0063067584764x_{23} = 44.0063067584764
x24=78.5514844556215x_{24} = 78.5514844556215
x25=94.2571139671403x_{25} = 94.2571139671403
x26=12.689444473502x_{26} = 12.689444473502
x27=56.5661872052363x_{27} = 56.5661872052363
x28=34.5901303273917x_{28} = 34.5901303273917
x29=91.1159155945839x_{29} = 91.1159155945839
x30=9.60585403951963x_{30} = 9.60585403951963
x31=28.3165366149909x_{31} = 28.3165366149909
x32=50.2857805952779x_{32} = 50.2857805952779
x33=15.7992708977347x_{33} = 15.7992708977347

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
[97.3983409430341,∞)\left[97.3983409430341, \infty\right)
Convex at the intervals
(−∞,3.79739269085901]\left(-\infty, 3.79739269085901\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)2sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\log{\left(x \right)}^{2} \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)2sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\log{\left(x \right)}^{2} \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 log(x)^2*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(x)2sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}^{2} \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)2sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}^{2} \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)2sin⁡(x)=−log⁡(−x)2sin⁡(x)\log{\left(x \right)}^{2} \sin{\left(x \right)} = - \log{\left(- x \right)}^{2} \sin{\left(x \right)}
- No
log⁡(x)2sin⁡(x)=log⁡(−x)2sin⁡(x)\log{\left(x \right)}^{2} \sin{\left(x \right)} = \log{\left(- x \right)}^{2} \sin{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator