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Plotting a function graph/ cosx*cos(2x)

Graphing y = cosx*cos(2x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cos(x)*cos(2*x)
f(x)=cos⁡(x)cos⁡(2x)f{\left(x \right)} = \cos{\left(x \right)} \cos{\left(2 x \right)} 
f = cos(x)*cos(2*x)
The graph of the function
02468-8-6-4-2-10102-2
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:
cos⁡(x)cos⁡(2x)=0\cos{\left(x \right)} \cos{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π2x_{1} = - \frac{\pi}{2}
x2=−π4x_{2} = - \frac{\pi}{4}
x3=π4x_{3} = \frac{\pi}{4}
x4=π2x_{4} = \frac{\pi}{2}
Numerical solution
x1=4.71238898038469x_{1} = 4.71238898038469
x2=−55.7632696012188x_{2} = -55.7632696012188
x3=7.85398163397448x_{3} = 7.85398163397448
x4=2.35619449019234x_{4} = 2.35619449019234
x5=−27.4889357189107x_{5} = -27.4889357189107
x6=99.7455667514759x_{6} = 99.7455667514759
x7=33.7721210260903x_{7} = 33.7721210260903
x8=−54.1924732744239x_{8} = -54.1924732744239
x9=38.484510006475x_{9} = 38.484510006475
x10=−32.2013246992954x_{10} = -32.2013246992954
x11=95.8185759344887x_{11} = 95.8185759344887
x12=−71.4712328691678x_{12} = -71.4712328691678
x13=−45.553093477052x_{13} = -45.553093477052
x14=26.7035375555132x_{14} = 26.7035375555132
x15=−41.6261026600648x_{15} = -41.6261026600648
x16=62.0464549083984x_{16} = 62.0464549083984
x17=91.8915851175014x_{17} = 91.8915851175014
x18=25.9181393921158x_{18} = 25.9181393921158
x19=−19.6349540849362x_{19} = -19.6349540849362
x20=−85.6083998103219x_{20} = -85.6083998103219
x21=−82.4668071567321x_{21} = -82.4668071567321
x22=24.3473430653209x_{22} = 24.3473430653209
x23=76.1836218495525x_{23} = 76.1836218495525
x24=48.6946861306418x_{24} = 48.6946861306418
x25=−58.1194640914112x_{25} = -58.1194640914112
x26=−49.4800842940392x_{26} = -49.4800842940392
x27=23.5619449019235x_{27} = 23.5619449019235
x28=82.4668071567321x_{28} = 82.4668071567321
x29=−80.1106126665397x_{29} = -80.1106126665397
x30=−93.4623814442964x_{30} = -93.4623814442964
x31=86.3937979737193x_{31} = 86.3937979737193
x32=−99.7455667514759x_{32} = -99.7455667514759
x33=45.553093477052x_{33} = 45.553093477052
x34=77.7544181763474x_{34} = 77.7544181763474
x35=51.8362787842316x_{35} = 51.8362787842316
x36=46.3384916404494x_{36} = 46.3384916404494
x37=−5.49778714378214x_{37} = -5.49778714378214
x38=−14.1371669411541x_{38} = -14.1371669411541
x39=16.4933614313464x_{39} = 16.4933614313464
x40=−11.7809724509617x_{40} = -11.7809724509617
x41=20.4203522483337x_{41} = 20.4203522483337
x42=−3.92699081698724x_{42} = -3.92699081698724
x43=−73.8274273593601x_{43} = -73.8274273593601
x44=−18.0641577581413x_{44} = -18.0641577581413
x45=−98.174770424681x_{45} = -98.174770424681
x46=−29.845130209103x_{46} = -29.845130209103
x47=−89.5353906273091x_{47} = -89.5353906273091
x48=64.4026493985908x_{48} = 64.4026493985908
x49=−62.0464549083984x_{49} = -62.0464549083984
x50=18.0641577581413x_{50} = 18.0641577581413
x51=29.845130209103x_{51} = 29.845130209103
x52=60.4756585816035x_{52} = 60.4756585816035
x53=11.7809724509617x_{53} = 11.7809724509617
x54=69.9004365423729x_{54} = 69.9004365423729
x55=−51.8362787842316x_{55} = -51.8362787842316
x56=92.6769832808989x_{56} = 92.6769832808989
x57=55.7632696012188x_{57} = 55.7632696012188
x58=47.9092879672443x_{58} = 47.9092879672443
x59=−63.6172512351933x_{59} = -63.6172512351933
x60=−67.5442420521806x_{60} = -67.5442420521806
x61=54.1924732744239x_{61} = 54.1924732744239
x62=−47.9092879672443x_{62} = -47.9092879672443
x63=−36.1283155162826x_{63} = -36.1283155162826
x64=−7.85398163397448x_{64} = -7.85398163397448
x65=14.1371669411541x_{65} = 14.1371669411541
x66=89.5353906273091x_{66} = 89.5353906273091
x67=40.0553063332699x_{67} = 40.0553063332699
x68=80.1106126665397x_{68} = 80.1106126665397
x69=−84.037603483527x_{69} = -84.037603483527
x70=−25.9181393921158x_{70} = -25.9181393921158
x71=−10.2101761241668x_{71} = -10.2101761241668
x72=42.4115008234622x_{72} = 42.4115008234622
x73=−91.8915851175014x_{73} = -91.8915851175014
x74=−69.9004365423729x_{74} = -69.9004365423729
x75=70.6858347057703x_{75} = 70.6858347057703
x76=36.1283155162826x_{76} = 36.1283155162826
x77=90.3207887907066x_{77} = 90.3207887907066
x78=−1.5707963267949x_{78} = -1.5707963267949
x79=−95.8185759344887x_{79} = -95.8185759344887
x80=−60.4756585816035x_{80} = -60.4756585816035
x81=−23.5619449019235x_{81} = -23.5619449019235
x82=3.92699081698724x_{82} = 3.92699081698724
x83=32.2013246992954x_{83} = 32.2013246992954
x84=−32.9867228626928x_{84} = -32.9867228626928
x85=67.5442420521806x_{85} = 67.5442420521806
x86=−77.7544181763474x_{86} = -77.7544181763474
x87=−17.2787595947439x_{87} = -17.2787595947439
x88=−76.1836218495525x_{88} = -76.1836218495525
x89=98.174770424681x_{89} = 98.174770424681
x90=−40.0553063332699x_{90} = -40.0553063332699
x91=73.8274273593601x_{91} = 73.8274273593601
x92=58.1194640914112x_{92} = 58.1194640914112
x93=1.5707963267949x_{93} = 1.5707963267949
x94=68.329640215578x_{94} = 68.329640215578
x95=−33.7721210260903x_{95} = -33.7721210260903
x96=10.2101761241668x_{96} = 10.2101761241668
x97=84.037603483527x_{97} = 84.037603483527
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)*cos(2*x).
cos⁡(0)cos⁡(0⋅2)\cos{\left(0 \right)} \cos{\left(0 \cdot 2 \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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)cos⁡(2x)−2sin⁡(2x)cos⁡(x)=0- \sin{\left(x \right)} \cos{\left(2 x \right)} - 2 \sin{\left(2 x \right)} \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
x3=i(log⁡(3)−log⁡(−2−5i))2x_{3} = \frac{i \left(\log{\left(3 \right)} - \log{\left(-2 - \sqrt{5} i \right)}\right)}{2}
x4=i(log⁡(3)−log⁡(−2+5i))2x_{4} = \frac{i \left(\log{\left(3 \right)} - \log{\left(-2 + \sqrt{5} i \right)}\right)}{2}
The values of the extrema at the points:
(0, 1)

(pi, -1)

   /     /         ___\         \                                           /  /     /         ___\         \\ 
 I*\- log\-2 - I*\/ 5 / + log(3)/     /  /     /         ___\         \\    |I*\- log\-2 - I*\/ 5 / + log(3)/| 
(--------------------------------, cos\I*\- log\-2 - I*\/ 5 / + log(3)//*cos|--------------------------------|)
                2                                                           \               2                / 

   /     /         ___\         \                                           /  /     /         ___\         \\ 
 I*\- log\-2 + I*\/ 5 / + log(3)/     /  /     /         ___\         \\    |I*\- log\-2 + I*\/ 5 / + log(3)/| 
(--------------------------------, cos\I*\- log\-2 + I*\/ 5 / + log(3)//*cos|--------------------------------|)
                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:
Minima of the function at points:
x1=πx_{1} = \pi
x2=−π2+atan⁡(52)2x_{2} = - \frac{\pi}{2} + \frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2}
x3=−atan⁡(52)2+π2x_{3} = - \frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} + \frac{\pi}{2}
Maxima of the function at points:
x3=0x_{3} = 0
Decreasing at intervals
[π,∞)\left[\pi, \infty\right)
Increasing at intervals
(−∞,−π2+atan⁡(52)2]∪[0,−atan⁡(52)2+π2]\left(-\infty, - \frac{\pi}{2} + \frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2}\right] \cup \left[0, - \frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} + \frac{\pi}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(cos⁡(x)cos⁡(2x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\cos{\left(x \right)} \cos{\left(2 x \right)}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞(cos⁡(x)cos⁡(2x))=⟨−1,1⟩\lim_{x \to \infty}\left(\cos{\left(x \right)} \cos{\left(2 x \right)}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)*cos(2*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x)cos⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \cos{\left(2 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x)cos⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \cos{\left(2 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:
cos⁡(x)cos⁡(2x)=cos⁡(x)cos⁡(2x)\cos{\left(x \right)} \cos{\left(2 x \right)} = \cos{\left(x \right)} \cos{\left(2 x \right)}
- Yes
cos⁡(x)cos⁡(2x)=−cos⁡(x)cos⁡(2x)\cos{\left(x \right)} \cos{\left(2 x \right)} = - \cos{\left(x \right)} \cos{\left(2 x \right)}
- No
so, the function
is
even
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