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  • How to use it?

  • Graphing y =:
  • x^4-2
  • x^3-x^3
  • x^3-x^2-2x
  • |x+3|-|x-1|+x+2
  • Limit of the function:
  • log(cos(x))/x^2 log(cos(x))/x^2

  • Identical expressions

  • log(cos(x))/x^ two
  • logarithm of ( co sinus of e of (x)) divide by x squared
  • logarithm of ( co sinus of e of (x)) divide by x to the power of two
  • log(cos(x))/x2
  • logcosx/x2
  • log(cos(x))/x²
  • log(cos(x))/x to the power of 2
  • logcosx/x^2
  • log(cos(x)) divide by x^2

  • Similar expressions

  • log(cosx)/x^2
Plotting a function graph/ log(cos(x))/x^2

Graphing y = log(cos(x))/x^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       log(cos(x))
f(x) = -----------
             2    
            x
f(x)=log⁡(cos⁡(x))x2f{\left(x \right)} = \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}} 
f = log(cos(x))/x^2
The graph of the function
02468-8-6-4-2-10102-2
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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⁡(cos⁡(x))x2=0\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=2πx_{1} = 2 \pi
Numerical solution
x1=62.8318527292118x_{1} = 62.8318527292118
x2=25.1327406153308x_{2} = 25.1327406153308
x3=−37.699115831174x_{3} = -37.699115831174
x4=18.8495566653649x_{4} = 18.8495566653649
x5=37.6991120390888x_{5} = 37.6991120390888
x6=−94.2477820859381x_{6} = -94.2477820859381
x7=−43.9822971744421x_{7} = -43.9822971744421
x8=25.1327418019752x_{8} = 25.1327418019752
x9=94.2477796642264x_{9} = 94.2477796642264
x10=18.8495568924594x_{10} = 18.8495568924594
x11=12.5663724671728x_{11} = 12.5663724671728
x12=12.566371085894x_{12} = 12.566371085894
x13=43.9822975717062x_{13} = 43.9822975717062
x14=−100.530964572061x_{14} = -100.530964572061
x15=−25.1327401011047x_{15} = -25.1327401011047
x16=−37.6991123409791x_{16} = -37.6991123409791
x17=43.9822969520563x_{17} = 43.9822969520563
x18=−37.699111877239x_{18} = -37.699111877239
x19=−18.8495552662988x_{19} = -18.8495552662988
x20=−31.4159262422395x_{20} = -31.4159262422395
x21=−87.9645944082851x_{21} = -87.9645944082851
x22=−62.8318524778695x_{22} = -62.8318524778695
x23=−87.9645943472471x_{23} = -87.9645943472471
x24=−50.2654827533158x_{24} = -50.2654827533158
x25=−18.8495564609393x_{25} = -18.8495564609393
x26=6.28318425041486x_{26} = 6.28318425041486
x27=−100.530965327794x_{27} = -100.530965327794
x28=87.9645940945969x_{28} = 87.9645940945969
x29=81.6814092014976x_{29} = 81.6814092014976
x30=56.5486675910765x_{30} = 56.5486675910765
x31=69.1150378100591x_{31} = 69.1150378100591
x32=−87.9645943582944x_{32} = -87.9645943582944
x33=50.2654825592132x_{33} = 50.2654825592132
x34=−94.2477799274769x_{34} = -94.2477799274769
x35=−62.8318552794767x_{35} = -62.8318552794767
x36=50.2654824463205x_{36} = 50.2654824463205
x37=62.8318541663282x_{37} = 62.8318541663282
x38=56.5486697853444x_{38} = 56.5486697853444
x39=−31.4159240473036x_{39} = -31.4159240473036
x40=56.5486680400228x_{40} = 56.5486680400228
x41=75.3982240657869x_{41} = 75.3982240657869
x42=−81.681409137626x_{42} = -81.681409137626
x43=31.4159254853307x_{43} = 31.4159254853307
x44=−94.2477794361694x_{44} = -94.2477794361694
x45=−69.1150387430457x_{45} = -69.1150387430457
x46=−25.1327415699717x_{46} = -25.1327415699717
x47=94.2477796093521x_{47} = 94.2477796093521
x48=−75.3982212667289x_{48} = -75.3982212667289
x49=−43.9822973028755x_{49} = -43.9822973028755
x50=6.28318528379982x_{50} = 6.28318528379982
x51=−56.5486674063126x_{51} = -56.5486674063126
x52=31.4159236692146x_{52} = 31.4159236692146
x53=−81.6814088468767x_{53} = -81.6814088468767
x54=81.6814067097954x_{54} = 81.6814067097954
x55=69.1150385422669x_{55} = 69.1150385422669
x56=100.530964751937x_{56} = 100.530964751937
x57=87.9645944170753x_{57} = 87.9645944170753
x58=−56.5486687958913x_{58} = -56.5486687958913
x59=6.28318536623994x_{59} = 6.28318536623994
x60=−31.4159267218315x_{60} = -31.4159267218315
x61=37.6991140183594x_{61} = 37.6991140183594
x62=87.9645943359894x_{62} = 87.9645943359894
x63=−75.3982238842683x_{63} = -75.3982238842683
x64=50.2654824267459x_{64} = 50.2654824267459
x65=100.530965170832x_{65} = 100.530965170832
x66=−12.5663711881532x_{66} = -12.5663711881532
x67=94.2477796240335x_{67} = 94.2477796240335
x68=31.4159268946166x_{68} = 31.4159268946166
x69=75.3982238631513x_{69} = 75.3982238631513
x70=−37.6991117133998x_{70} = -37.6991117133998
x71=37.6991114226912x_{71} = 37.6991114226912
x72=62.8318534194081x_{72} = 62.8318534194081
x73=12.5663704222821x_{73} = 12.5663704222821
x74=69.1150389965986x_{74} = 69.1150389965986
x75=−50.2654848480677x_{75} = -50.2654848480677
x76=−69.115038371213x_{76} = -69.115038371213
x77=−12.5663702114039x_{77} = -12.5663702114039
x78=−25.1327411327427x_{78} = -25.1327411327427
x79=−81.6814090383396x_{79} = -81.6814090383396
x80=37.6991093650365x_{80} = 37.6991093650365
x81=43.9823007506628x_{81} = 43.9823007506628
x82=−6.28318703759797x_{82} = -6.28318703759797
x83=−6.28318509127283x_{83} = -6.28318509127283
x84=−75.3982237850615x_{84} = -75.3982237850615
x85=18.8495555495816x_{85} = 18.8495555495816
x86=−69.1150373549062x_{86} = -69.1150373549062
x87=43.9822971694655x_{87} = 43.9822971694655
x88=−56.5486680686757x_{88} = -56.5486680686757
x89=−62.8318536636829x_{89} = -62.8318536636829
x90=−6.28318500859429x_{90} = -6.28318500859429
x91=−43.9822971370899x_{91} = -43.9822971370899
x92=75.3982227157249x_{92} = 75.3982227157249
x93=−50.2654822745065x_{93} = -50.2654822745065
x94=81.6814088355208x_{94} = 81.6814088355208
x95=−12.5663714894404x_{95} = -12.5663714894404
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(cos(x))/x^2.
log⁡(cos⁡(0))02\frac{\log{\left(\cos{\left(0 \right)} \right)}}{0^{2}}
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
−sin⁡(x)x2cos⁡(x)−2log⁡(cos⁡(x))x3=0- \frac{\sin{\left(x \right)}}{x^{2} \cos{\left(x \right)}} - \frac{2 \log{\left(\cos{\left(x \right)} \right)}}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=43.9822971502571x_{1} = 43.9822971502571
x2=−43.9822971502571x_{2} = -43.9822971502571
x3=81.6814089933346x_{3} = 81.6814089933346
x4=−31.4159265358979x_{4} = -31.4159265358979
x5=−25.1327412287183x_{5} = -25.1327412287183
x6=−94.2477796076938x_{6} = -94.2477796076938
x7=6.28318530717959x_{7} = 6.28318530717959
x8=−50.2654824574367x_{8} = -50.2654824574367
x9=−75.398223686155x_{9} = -75.398223686155
x10=−56.5486677646163x_{10} = -56.5486677646163
x11=50.2654824574367x_{11} = 50.2654824574367
x12=−69.1150383789755x_{12} = -69.1150383789755
x13=−100.530964914873x_{13} = -100.530964914873
x14=56.5486677646163x_{14} = 56.5486677646163
x15=−62.8318530717959x_{15} = -62.8318530717959
x16=−87.9645943005142x_{16} = -87.9645943005142
x17=18.8495559215388x_{17} = 18.8495559215388
x18=100.530964914873x_{18} = 100.530964914873
x19=62.8318530717959x_{19} = 62.8318530717959
x20=94.2477796076938x_{20} = 94.2477796076938
x21=12.5663706143592x_{21} = 12.5663706143592
x22=87.9645943005142x_{22} = 87.9645943005142
x23=69.1150383789755x_{23} = 69.1150383789755
x24=−6.28318530717959x_{24} = -6.28318530717959
x25=75.398223686155x_{25} = 75.398223686155
x26=−37.6991118430775x_{26} = -37.6991118430775
x27=−12.5663706143592x_{27} = -12.5663706143592
x28=−18.8495559215388x_{28} = -18.8495559215388
x29=31.4159265358979x_{29} = 31.4159265358979
x30=−81.6814089933346x_{30} = -81.6814089933346
x31=37.6991118430775x_{31} = 37.6991118430775
x32=25.1327412287183x_{32} = 25.1327412287183
The values of the extrema at the points:
(43.982297150257104, 0)

(-43.982297150257104, 0)

(81.68140899333463, 0)

(-31.41592653589793, 0)

(-25.132741228718345, 0)

(-94.2477796076938, 0)

(6.283185307179586, 0)

(-50.26548245743669, 0)

(-75.39822368615503, 0)

(-56.548667764616276, 0)

(50.26548245743669, 0)

(-69.11503837897546, 0)

(-100.53096491487338, 0)

(56.548667764616276, 0)

(-62.83185307179586, 0)

(-87.96459430051421, 0)

(18.84955592153876, 0)

(100.53096491487338, 0)

(62.83185307179586, 0)

(94.2477796076938, 0)

(12.566370614359172, 0)

(87.96459430051421, 0)

(69.11503837897546, 0)

(-6.283185307179586, 0)

(75.39822368615503, 0)

(-37.69911184307752, 0)

(-12.566370614359172, 0)

(-18.84955592153876, 0)

(31.41592653589793, 0)

(-81.68140899333463, 0)

(37.69911184307752, 0)

(25.132741228718345, 0)


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:
The function has no minima
Maxima of the function at points:
x32=43.9822971502571x_{32} = 43.9822971502571
x32=−43.9822971502571x_{32} = -43.9822971502571
x32=81.6814089933346x_{32} = 81.6814089933346
x32=−31.4159265358979x_{32} = -31.4159265358979
x32=−25.1327412287183x_{32} = -25.1327412287183
x32=−94.2477796076938x_{32} = -94.2477796076938
x32=6.28318530717959x_{32} = 6.28318530717959
x32=−50.2654824574367x_{32} = -50.2654824574367
x32=−75.398223686155x_{32} = -75.398223686155
x32=−56.5486677646163x_{32} = -56.5486677646163
x32=50.2654824574367x_{32} = 50.2654824574367
x32=−69.1150383789755x_{32} = -69.1150383789755
x32=−100.530964914873x_{32} = -100.530964914873
x32=56.5486677646163x_{32} = 56.5486677646163
x32=−62.8318530717959x_{32} = -62.8318530717959
x32=−87.9645943005142x_{32} = -87.9645943005142
x32=18.8495559215388x_{32} = 18.8495559215388
x32=100.530964914873x_{32} = 100.530964914873
x32=62.8318530717959x_{32} = 62.8318530717959
x32=94.2477796076938x_{32} = 94.2477796076938
x32=12.5663706143592x_{32} = 12.5663706143592
x32=87.9645943005142x_{32} = 87.9645943005142
x32=69.1150383789755x_{32} = 69.1150383789755
x32=−6.28318530717959x_{32} = -6.28318530717959
x32=75.398223686155x_{32} = 75.398223686155
x32=−37.6991118430775x_{32} = -37.6991118430775
x32=−12.5663706143592x_{32} = -12.5663706143592
x32=−18.8495559215388x_{32} = -18.8495559215388
x32=31.4159265358979x_{32} = 31.4159265358979
x32=−81.6814089933346x_{32} = -81.6814089933346
x32=37.6991118430775x_{32} = 37.6991118430775
x32=25.1327412287183x_{32} = 25.1327412287183
Decreasing at intervals
(−∞,−100.530964914873]\left(-\infty, -100.530964914873\right]
Increasing at intervals
[100.530964914873,∞)\left[100.530964914873, \infty\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
−sin⁡2(x)cos⁡2(x)−1+4sin⁡(x)xcos⁡(x)+6log⁡(cos⁡(x))x2x2=0\frac{- \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 1 + \frac{4 \sin{\left(x \right)}}{x \cos{\left(x \right)}} + \frac{6 \log{\left(\cos{\left(x \right)} \right)}}{x^{2}}}{x^{2}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(log⁡(cos⁡(x))x2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(log⁡(cos⁡(x))x2)=0\lim_{x \to \infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(cos(x))/x^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(cos⁡(x))xx2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x x^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(cos⁡(x))xx2)=0\lim_{x \to \infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x x^{2}}\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⁡(cos⁡(x))x2=log⁡(cos⁡(x))x2\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}} = \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}
- Yes
log⁡(cos⁡(x))x2=−log⁡(cos⁡(x))x2\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}} = - \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}
- No
so, the function
is
even
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