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Plotting a function graph/ ln(sqrt(2*cos(x)))

Graphing y = ln(sqrt(2*cos(x)))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /  __________\
f(x) = log\\/ 2*cos(x) /
f(x)=log⁡(2cos⁡(x))f{\left(x \right)} = \log{\left(\sqrt{2 \cos{\left(x \right)}} \right)} 
f = log(sqrt(2*cos(x)))
The graph of the function
02468-8-6-4-2-10102.5-2.5
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⁡(2cos⁡(x))=0\log{\left(\sqrt{2 \cos{\left(x \right)}} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π3x_{1} = \frac{\pi}{3}
x2=5π3x_{2} = \frac{5 \pi}{3}
Numerical solution
x1=30.3687289847013x_{1} = 30.3687289847013
x2=−5.23598775598299x_{2} = -5.23598775598299
x3=76.4454212373516x_{3} = 76.4454212373516
x4=32.4631240870945x_{4} = 32.4631240870945
x5=74.3510261349584x_{5} = 74.3510261349584
x6=−95.2949771588904x_{6} = -95.2949771588904
x7=−82.7286065445312x_{7} = -82.7286065445312
x8=80.634211442138x_{8} = 80.634211442138
x9=13.6135681655558x_{9} = 13.6135681655558
x10=−76.4454212373516x_{10} = -76.4454212373516
x11=24.0855436775217x_{11} = 24.0855436775217
x12=−55.5014702134197x_{12} = -55.5014702134197
x13=−7.33038285837618x_{13} = -7.33038285837618
x14=−61.7846555205993x_{14} = -61.7846555205993
x15=86.9173967493176x_{15} = 86.9173967493176
x16=68.0678408277789x_{16} = 68.0678408277789
x17=57.5958653158129x_{17} = 57.5958653158129
x18=−101.57816246607x_{18} = -101.57816246607
x19=70.162235930172x_{19} = 70.162235930172
x20=−30.3687289847013x_{20} = -30.3687289847013
x21=−51.3126800086333x_{21} = -51.3126800086333
x22=101.57816246607x_{22} = 101.57816246607
x23=11.5191730631626x_{23} = 11.5191730631626
x24=93.2005820564972x_{24} = 93.2005820564972
x25=−17.8023583703422x_{25} = -17.8023583703422
x26=−233.525053916841x_{26} = -233.525053916841
x27=89.0117918517108x_{27} = 89.0117918517108
x28=−49.2182849062401x_{28} = -49.2182849062401
x29=−24.0855436775217x_{29} = -24.0855436775217
x30=−68.0678408277789x_{30} = -68.0678408277789
x31=−80.634211442138x_{31} = -80.634211442138
x32=−11.5191730631626x_{32} = -11.5191730631626
x33=38.7463093942741x_{33} = 38.7463093942741
x34=−86.9173967493176x_{34} = -86.9173967493176
x35=−74.3510261349584x_{35} = -74.3510261349584
x36=42.9350995990605x_{36} = 42.9350995990605
x37=36.6519142918809x_{37} = 36.6519142918809
x38=49.2182849062401x_{38} = 49.2182849062401
x39=−19.8967534727354x_{39} = -19.8967534727354
x40=5.23598775598299x_{40} = 5.23598775598299
x41=61.7846555205993x_{41} = 61.7846555205993
x42=−57.5958653158129x_{42} = -57.5958653158129
x43=−26.1799387799149x_{43} = -26.1799387799149
x44=−70.162235930172x_{44} = -70.162235930172
x45=−63.8790506229925x_{45} = -63.8790506229925
x46=17.8023583703422x_{46} = 17.8023583703422
x47=55.5014702134197x_{47} = 55.5014702134197
x48=−32.4631240870945x_{48} = -32.4631240870945
x49=−13.6135681655558x_{49} = -13.6135681655558
x50=99.4837673636768x_{50} = 99.4837673636768
x51=82.7286065445312x_{51} = 82.7286065445312
x52=63.8790506229925x_{52} = 63.8790506229925
x53=45.0294947014537x_{53} = 45.0294947014537
x54=−99.4837673636768x_{54} = -99.4837673636768
x55=19.8967534727354x_{55} = 19.8967534727354
x56=−93.2005820564972x_{56} = -93.2005820564972
x57=26.1799387799149x_{57} = 26.1799387799149
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(sqrt(2*cos(x))).
log⁡(2cos⁡(0))\log{\left(\sqrt{2 \cos{\left(0 \right)}} \right)}
The result:
f(0)=log⁡(2)f{\left(0 \right)} = \log{\left(\sqrt{2} \right)}
The point:
(0, log(sqrt(2)))
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)2cos⁡(x)=0- \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
The values of the extrema at the points:
       /  ___\ 
(0, log\\/ 2 /)

     pi*I      /  ___\ 
(pi, ---- + log\\/ 2 /)
      2


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:
x2=0x_{2} = 0
Decreasing at intervals
(−∞,0]\left(-\infty, 0\right]
Increasing at intervals
[0,∞)\left[0, \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)+12=0- \frac{\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1}{2} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞log⁡(2cos⁡(x))=log⁡(2⟨0,1⟩)\lim_{x \to -\infty} \log{\left(\sqrt{2 \cos{\left(x \right)}} \right)} = \log{\left(\sqrt{2} \left\langle 0, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=log⁡(2⟨0,1⟩)y = \log{\left(\sqrt{2} \left\langle 0, 1\right\rangle \right)}
lim⁡x→∞log⁡(2cos⁡(x))=log⁡(2⟨0,1⟩)\lim_{x \to \infty} \log{\left(\sqrt{2 \cos{\left(x \right)}} \right)} = \log{\left(\sqrt{2} \left\langle 0, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=log⁡(2⟨0,1⟩)y = \log{\left(\sqrt{2} \left\langle 0, 1\right\rangle \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(sqrt(2*cos(x))), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(2cos⁡(x))x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\sqrt{2 \cos{\left(x \right)}} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(2cos⁡(x))x)=0\lim_{x \to \infty}\left(\frac{\log{\left(\sqrt{2 \cos{\left(x \right)}} \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⁡(2cos⁡(x))=log⁡(2cos⁡(x))\log{\left(\sqrt{2 \cos{\left(x \right)}} \right)} = \log{\left(\sqrt{2 \cos{\left(x \right)}} \right)}
- Yes
log⁡(2cos⁡(x))=−log⁡(2cos⁡(x))\log{\left(\sqrt{2 \cos{\left(x \right)}} \right)} = - \log{\left(\sqrt{2 \cos{\left(x \right)}} \right)}
- No
so, the function
is
even
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