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cosx-1/2cos2x
Plotting a function graph/ cosx-1/2cos2x

Graphing y = cosx-1/2cos2x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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                cos(2*x)
f(x) = cos(x) - --------
                   2
f(x)=cos⁡(x)−cos⁡(2x)2f{\left(x \right)} = \cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{2} 
f = cos(x) - cos(2*x)/2
The graph of the function
0-70-60-50-40-30-20-10102030405060702.5-2.5
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:
cos⁡(x)−cos⁡(2x)2=0\cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=i(log⁡(2)−log⁡(−3+1−2⋅34i))x_{1} = i \left(\log{\left(2 \right)} - \log{\left(- \sqrt{3} + 1 - \sqrt{2} \cdot \sqrt[4]{3} i \right)}\right)
x2=i(log⁡(2)−log⁡(−3+1+2⋅34i))x_{2} = i \left(\log{\left(2 \right)} - \log{\left(- \sqrt{3} + 1 + \sqrt{2} \cdot \sqrt[4]{3} i \right)}\right)
Numerical solution
x1=−77.3437544456587x_{1} = -77.3437544456587
x2=71.0605691384791x_{2} = 71.0605691384791
x3=42.0367663907535x_{3} = 42.0367663907535
x4=64.7773838312995x_{4} = 64.7773838312995
x5=−79.735878233831x_{5} = -79.735878233831
x6=48.3199516979331x_{6} = 48.3199516979331
x7=−8.22871606668322x_{7} = -8.22871606668322
x8=−20.7950866810424x_{8} = -20.7950866810424
x9=−42.0367663907535x_{9} = -42.0367663907535
x10=54.6031370051126x_{10} = 54.6031370051126
x11=−92.3022488481902x_{11} = -92.3022488481902
x12=−130.001360691268x_{12} = -130.001360691268
x13=20.7950866810424x_{13} = 20.7950866810424
x14=−10.6208398548555x_{14} = -10.6208398548555
x15=−73.4526929266514x_{15} = -73.4526929266514
x16=−67.1695076194718x_{16} = -67.1695076194718
x17=4324.77702209906x_{17} = 4324.77702209906
x18=14.5119013738628x_{18} = 14.5119013738628
x19=−89.9101250600178x_{19} = -89.9101250600178
x20=89.9101250600178x_{20} = 89.9101250600178
x21=73.4526929266514x_{21} = 73.4526929266514
x22=−96.1933103671974x_{22} = -96.1933103671974
x23=98.5854341553698x_{23} = 98.5854341553698
x24=−98.5854341553698x_{24} = -98.5854341553698
x25=−14.5119013738628x_{25} = -14.5119013738628
x26=−23.1872104692147x_{26} = -23.1872104692147
x27=39.6446426025812x_{27} = 39.6446426025812
x28=−71.0605691384791x_{28} = -71.0605691384791
x29=10.6208398548555x_{29} = 10.6208398548555
x30=−35.7535810835739x_{30} = -35.7535810835739
x31=8.22871606668322x_{31} = 8.22871606668322
x32=79.735878233831x_{32} = 79.735878233831
x33=−52.2110132169403x_{33} = -52.2110132169403
x34=−29.4703957763943x_{34} = -29.4703957763943
x35=35.7535810835739x_{35} = 35.7535810835739
x36=92.3022488481902x_{36} = 92.3022488481902
x37=33.3614572954016x_{37} = 33.3614572954016
x38=1.94553075950364x_{38} = 1.94553075950364
x39=16.9040251620351x_{39} = 16.9040251620351
x40=29.4703957763943x_{40} = 29.4703957763943
x41=−86.0190635410106x_{41} = -86.0190635410106
x42=−39.6446426025812x_{42} = -39.6446426025812
x43=60.8863223122922x_{43} = 60.8863223122922
x44=−48.3199516979331x_{44} = -48.3199516979331
x45=45.9278279097607x_{45} = 45.9278279097607
x46=−45.9278279097607x_{46} = -45.9278279097607
x47=58.4941985241199x_{47} = 58.4941985241199
x48=96.1933103671974x_{48} = 96.1933103671974
x49=4.33765454767595x_{49} = 4.33765454767595
x50=−27.078271988222x_{50} = -27.078271988222
x51=77.3437544456587x_{51} = 77.3437544456587
x52=−4.33765454767595x_{52} = -4.33765454767595
x53=−58.4941985241199x_{53} = -58.4941985241199
x54=−33.3614572954016x_{54} = -33.3614572954016
x55=−1.94553075950364x_{55} = -1.94553075950364
x56=86.0190635410106x_{56} = 86.0190635410106
x57=52.2110132169403x_{57} = 52.2110132169403
x58=−83.6269397528383x_{58} = -83.6269397528383
x59=−54.6031370051126x_{59} = -54.6031370051126
x60=83.6269397528383x_{60} = 83.6269397528383
x61=−60.8863223122922x_{61} = -60.8863223122922
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)/2.
−cos⁡(2⋅0)2+cos⁡(0)- \frac{\cos{\left(2 \cdot 0 \right)}}{2} + \cos{\left(0 \right)}
The result:
f(0)=12f{\left(0 \right)} = \frac{1}{2}
The point:
(0, 1/2)
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)+sin⁡(2x)=0- \sin{\left(x \right)} + \sin{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−5π3x_{2} = - \frac{5 \pi}{3}
x3=−πx_{3} = - \pi
x4=−π3x_{4} = - \frac{\pi}{3}
x5=π3x_{5} = \frac{\pi}{3}
x6=πx_{6} = \pi
x7=5π3x_{7} = \frac{5 \pi}{3}
The values of the extrema at the points:
(0, 1/2)

 -5*pi       
(------, 3/4)
   3         

(-pi, -3/2)

 -pi       
(----, 3/4)
  3        

 pi      
(--, 3/4)
 3       

(pi, -3/2)

 5*pi      
(----, 3/4)
  3


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=0x_{1} = 0
x2=−πx_{2} = - \pi
x3=πx_{3} = \pi
Maxima of the function at points:
x3=−5π3x_{3} = - \frac{5 \pi}{3}
x3=−π3x_{3} = - \frac{\pi}{3}
x3=π3x_{3} = \frac{\pi}{3}
x3=5π3x_{3} = \frac{5 \pi}{3}
Decreasing at intervals
[π,∞)\left[\pi, \infty\right)
Increasing at intervals
(−∞,−π]\left(-\infty, - \pi\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
−cos⁡(x)+2cos⁡(2x)=0- \cos{\left(x \right)} + 2 \cos{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=−ilog⁡(18+338−2i−33+158)x_{1} = - i \log{\left(\frac{1}{8} + \frac{\sqrt{33}}{8} - \frac{\sqrt{2} i \sqrt{- \sqrt{33} + 15}}{8} \right)}
x2=−ilog⁡(18+338+2i−33+158)x_{2} = - i \log{\left(\frac{1}{8} + \frac{\sqrt{33}}{8} + \frac{\sqrt{2} i \sqrt{- \sqrt{33} + 15}}{8} \right)}
x3=−ilog⁡(−338+18−2i33+158)x_{3} = - i \log{\left(- \frac{\sqrt{33}}{8} + \frac{1}{8} - \frac{\sqrt{2} i \sqrt{\sqrt{33} + 15}}{8} \right)}
x4=−ilog⁡(−338+18+2i33+158)x_{4} = - i \log{\left(- \frac{\sqrt{33}}{8} + \frac{1}{8} + \frac{\sqrt{2} i \sqrt{\sqrt{33} + 15}}{8} \right)}

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
[atan⁡(233+15−33+1)+π,∞)\left[\operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{\sqrt{33} + 15}}{- \sqrt{33} + 1} \right)} + \pi, \infty\right)
Convex at the intervals
(−∞,−atan⁡(2−33+151+33)]∪[atan⁡(2−33+151+33),atan⁡(233+15−33+1)+π]\left(-\infty, - \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- \sqrt{33} + 15}}{1 + \sqrt{33}} \right)}\right] \cup \left[\operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{- \sqrt{33} + 15}}{1 + \sqrt{33}} \right)}, \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{\sqrt{33} + 15}}{- \sqrt{33} + 1} \right)} + \pi\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)2)=⟨−32,32⟩\lim_{x \to -\infty}\left(\cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{2}\right) = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−32,32⟩y = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
lim⁡x→∞(cos⁡(x)−cos⁡(2x)2)=⟨−32,32⟩\lim_{x \to \infty}\left(\cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{2}\right) = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−32,32⟩y = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x) - cos(2*x)/2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x)−cos⁡(2x)2x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{2}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x)−cos⁡(2x)2x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{2}}{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)2=cos⁡(x)−cos⁡(2x)2\cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{2} = \cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{2}
- Yes
cos⁡(x)−cos⁡(2x)2=−cos⁡(x)+cos⁡(2x)2\cos{\left(x \right)} - \frac{\cos{\left(2 x \right)}}{2} = - \cos{\left(x \right)} + \frac{\cos{\left(2 x \right)}}{2}
- No
so, the function
is
even
The graph
Graphing y = cosx-1/2cos2x
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