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Plotting a function graph/ 1/2*cosx/3

Graphing y = 1/2*cosx/3

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       /cos(x)\
       |------|
       \  2   /
f(x) = --------
          3
f(x)=12cos⁡(x)3f{\left(x \right)} = \frac{\frac{1}{2} \cos{\left(x \right)}}{3} 
f = (cos(x)/2)/3
The graph of the function
02468-8-6-4-2-10100.5-0.5
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:
12cos⁡(x)3=0\frac{\frac{1}{2} \cos{\left(x \right)}}{3} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=42.4115008234622x_{1} = 42.4115008234622
x2=54.9778714378214x_{2} = 54.9778714378214
x3=−86.3937979737193x_{3} = -86.3937979737193
x4=−98.9601685880785x_{4} = -98.9601685880785
x5=29.845130209103x_{5} = 29.845130209103
x6=−42.4115008234622x_{6} = -42.4115008234622
x7=89.5353906273091x_{7} = 89.5353906273091
x8=−95.8185759344887x_{8} = -95.8185759344887
x9=−64.4026493985908x_{9} = -64.4026493985908
x10=14.1371669411541x_{10} = 14.1371669411541
x11=−17.2787595947439x_{11} = -17.2787595947439
x12=48.6946861306418x_{12} = 48.6946861306418
x13=−48.6946861306418x_{13} = -48.6946861306418
x14=−3623.82712591583x_{14} = -3623.82712591583
x15=−67.5442420521806x_{15} = -67.5442420521806
x16=−32.9867228626928x_{16} = -32.9867228626928
x17=−80.1106126665397x_{17} = -80.1106126665397
x18=83.2522053201295x_{18} = 83.2522053201295
x19=1.5707963267949x_{19} = 1.5707963267949
x20=10.9955742875643x_{20} = 10.9955742875643
x21=−7.85398163397448x_{21} = -7.85398163397448
x22=−76.9690200129499x_{22} = -76.9690200129499
x23=98.9601685880785x_{23} = 98.9601685880785
x24=−4.71238898038469x_{24} = -4.71238898038469
x25=−2266.65909956504x_{25} = -2266.65909956504
x26=36.1283155162826x_{26} = 36.1283155162826
x27=20.4203522483337x_{27} = 20.4203522483337
x28=23.5619449019235x_{28} = 23.5619449019235
x29=51.8362787842316x_{29} = 51.8362787842316
x30=−387.986692718339x_{30} = -387.986692718339
x31=−45.553093477052x_{31} = -45.553093477052
x32=45.553093477052x_{32} = 45.553093477052
x33=−1.5707963267949x_{33} = -1.5707963267949
x34=−10.9955742875643x_{34} = -10.9955742875643
x35=26.7035375555132x_{35} = 26.7035375555132
x36=67.5442420521806x_{36} = 67.5442420521806
x37=92.6769832808989x_{37} = 92.6769832808989
x38=−58.1194640914112x_{38} = -58.1194640914112
x39=73.8274273593601x_{39} = 73.8274273593601
x40=−39.2699081698724x_{40} = -39.2699081698724
x41=95.8185759344887x_{41} = 95.8185759344887
x42=−23.5619449019235x_{42} = -23.5619449019235
x43=−70.6858347057703x_{43} = -70.6858347057703
x44=80.1106126665397x_{44} = 80.1106126665397
x45=58.1194640914112x_{45} = 58.1194640914112
x46=−14.1371669411541x_{46} = -14.1371669411541
x47=32.9867228626928x_{47} = 32.9867228626928
x48=−83.2522053201295x_{48} = -83.2522053201295
x49=7.85398163397448x_{49} = 7.85398163397448
x50=−89.5353906273091x_{50} = -89.5353906273091
x51=−29.845130209103x_{51} = -29.845130209103
x52=76.9690200129499x_{52} = 76.9690200129499
x53=86.3937979737193x_{53} = 86.3937979737193
x54=70.6858347057703x_{54} = 70.6858347057703
x55=−26.7035375555132x_{55} = -26.7035375555132
x56=−36.1283155162826x_{56} = -36.1283155162826
x57=−92.6769832808989x_{57} = -92.6769832808989
x58=−51.8362787842316x_{58} = -51.8362787842316
x59=−73.8274273593601x_{59} = -73.8274273593601
x60=17.2787595947439x_{60} = 17.2787595947439
x61=64.4026493985908x_{61} = 64.4026493985908
x62=−20.4203522483337x_{62} = -20.4203522483337
x63=4.71238898038469x_{63} = 4.71238898038469
x64=−54.9778714378214x_{64} = -54.9778714378214
x65=39.2699081698724x_{65} = 39.2699081698724
x66=−61.261056745001x_{66} = -61.261056745001
x67=61.261056745001x_{67} = 61.261056745001
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (cos(x)/2)/3.
12cos⁡(0)3\frac{\frac{1}{2} \cos{\left(0 \right)}}{3}
The result:
f(0)=16f{\left(0 \right)} = \frac{1}{6}
The point:
(0, 1/6)
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)6=0- \frac{\sin{\left(x \right)}}{6} = 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, 1/6)

(pi, -1/6)


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
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\right)
Increasing at intervals
[0,π]\left[0, \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)6=0- \frac{\cos{\left(x \right)}}{6} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}

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
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Convex at the intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(12cos⁡(x)3)=⟨−16,16⟩\lim_{x \to -\infty}\left(\frac{\frac{1}{2} \cos{\left(x \right)}}{3}\right) = \left\langle - \frac{1}{6}, \frac{1}{6}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−16,16⟩y = \left\langle - \frac{1}{6}, \frac{1}{6}\right\rangle
lim⁡x→∞(12cos⁡(x)3)=⟨−16,16⟩\lim_{x \to \infty}\left(\frac{\frac{1}{2} \cos{\left(x \right)}}{3}\right) = \left\langle - \frac{1}{6}, \frac{1}{6}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−16,16⟩y = \left\langle - \frac{1}{6}, \frac{1}{6}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (cos(x)/2)/3, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x)6x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{6 x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x)6x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{6 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:
12cos⁡(x)3=12cos⁡(x)3\frac{\frac{1}{2} \cos{\left(x \right)}}{3} = \frac{\frac{1}{2} \cos{\left(x \right)}}{3}
- Yes
12cos⁡(x)3=−12cos⁡(x)3\frac{\frac{1}{2} \cos{\left(x \right)}}{3} = - \frac{\frac{1}{2} \cos{\left(x \right)}}{3}
- No
so, the function
is
even
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