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Plotting a function graph/ cot(x)-6*cos(x)

Graphing y = cot(x)-6*cos(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = cot(x) - 6*cos(x)
f(x)=−6cos⁡(x)+cot⁡(x)f{\left(x \right)} = - 6 \cos{\left(x \right)} + \cot{\left(x \right)} 
f = -6*cos(x) + cot(x)
The graph of the function
02468-8-6-4-2-1010-10001000
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:
−6cos⁡(x)+cot⁡(x)=0- 6 \cos{\left(x \right)} + \cot{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
x3=i(log⁡(6)−log⁡(−35+i))x_{3} = i \left(\log{\left(6 \right)} - \log{\left(- \sqrt{35} + i \right)}\right)
x4=i(log⁡(6)−log⁡(35+i))x_{4} = i \left(\log{\left(6 \right)} - \log{\left(\sqrt{35} + i \right)}\right)
Numerical solution
x1=17.2787595947439x_{1} = 17.2787595947439
x2=−23.5619449019235x_{2} = -23.5619449019235
x3=−87.7971462212945x_{3} = -87.7971462212945
x4=−43.8148490710374x_{4} = -43.8148490710374
x5=−14.1371669411541x_{5} = -14.1371669411541
x6=51.8362787842316x_{6} = 51.8362787842316
x7=−17.2787595947439x_{7} = -17.2787595947439
x8=−10.9955742875643x_{8} = -10.9955742875643
x9=−36.1283155162826x_{9} = -36.1283155162826
x10=−15.8754113471687x_{10} = -15.8754113471687
x11=−95.8185759344887x_{11} = -95.8185759344887
x12=−48.6946861306418x_{12} = -48.6946861306418
x13=−26.7035375555132x_{13} = -26.7035375555132
x14=4.71238898038469x_{14} = 4.71238898038469
x15=26.7035375555132x_{15} = 26.7035375555132
x16=89.5353906273091x_{16} = 89.5353906273091
x17=23.5619449019235x_{17} = 23.5619449019235
x18=14.1371669411541x_{18} = 14.1371669411541
x19=72.0891829533456x_{19} = 72.0891829533456
x20=95.8185759344887x_{20} = 95.8185759344887
x21=−61.261056745001x_{21} = -61.261056745001
x22=42.4115008234622x_{22} = 42.4115008234622
x23=58.1194640914112x_{23} = 58.1194640914112
x24=36.1283155162826x_{24} = 36.1283155162826
x25=29.845130209103x_{25} = 29.845130209103
x26=−73.8274273593601x_{26} = -73.8274273593601
x27=61.261056745001x_{27} = 61.261056745001
x28=48.6946861306418x_{28} = 48.6946861306418
x29=−4.71238898038469x_{29} = -4.71238898038469
x30=70.6858347057703x_{30} = 70.6858347057703
x31=−7.85398163397448x_{31} = -7.85398163397448
x32=−51.8362787842316x_{32} = -51.8362787842316
x33=−76.9690200129499x_{33} = -76.9690200129499
x34=−89.5353906273091x_{34} = -89.5353906273091
x35=−39.2699081698724x_{35} = -39.2699081698724
x36=80.1106126665397x_{36} = 80.1106126665397
x37=−42.4115008234622x_{37} = -42.4115008234622
x38=83.2522053201295x_{38} = 83.2522053201295
x39=−92.6769832808989x_{39} = -92.6769832808989
x40=28.1068858030884x_{40} = 28.1068858030884
x41=32.9867228626928x_{41} = 32.9867228626928
x42=45.553093477052x_{42} = 45.553093477052
x43=−59.8577084974258x_{43} = -59.8577084974258
x44=20.4203522483337x_{44} = 20.4203522483337
x45=64.4026493985908x_{45} = 64.4026493985908
x46=−32.9867228626928x_{46} = -32.9867228626928
x47=67.5442420521806x_{47} = 67.5442420521806
x48=−20.4203522483337x_{48} = -20.4203522483337
x49=−80.1106126665397x_{49} = -80.1106126665397
x50=7.85398163397448x_{50} = 7.85398163397448
x51=88.1320423797339x_{51} = 88.1320423797339
x52=−45.553093477052x_{52} = -45.553093477052
x53=76.9690200129499x_{53} = 76.9690200129499
x54=−1.5707963267949x_{54} = -1.5707963267949
x55=39.2699081698724x_{55} = 39.2699081698724
x56=−70.6858347057703x_{56} = -70.6858347057703
x57=−67.5442420521806x_{57} = -67.5442420521806
x58=−98.9601685880785x_{58} = -98.9601685880785
x59=−105.243353895258x_{59} = -105.243353895258
x60=−83.2522053201295x_{60} = -83.2522053201295
x61=−29.845130209103x_{61} = -29.845130209103
x62=−86.3937979737193x_{62} = -86.3937979737193
x63=98.9601685880785x_{63} = 98.9601685880785
x64=34.390071110268x_{64} = 34.390071110268
x65=73.8274273593601x_{65} = 73.8274273593601
x66=−58.1194640914112x_{66} = -58.1194640914112
x67=92.6769832808989x_{67} = 92.6769832808989
x68=54.9778714378214x_{68} = 54.9778714378214
x69=44.1497452294768x_{69} = 44.1497452294768
x70=86.3937979737193x_{70} = 86.3937979737193
x71=1.5707963267949x_{71} = 1.5707963267949
x72=−54.9778714378214x_{72} = -54.9778714378214
x73=−64.4026493985908x_{73} = -64.4026493985908
x74=10.9955742875643x_{74} = 10.9955742875643
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
6sin⁡(x)−cot⁡2(x)−1=06 \sin{\left(x \right)} - \cot^{2}{\left(x \right)} - 1 = 0
Solve this equation
Solutions are not found,
function may have no extrema
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
2((cot⁡2(x)+1)cot⁡(x)+3cos⁡(x))=02 \left(\left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + 3 \cos{\left(x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\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,π2]\left[- \frac{\pi}{2}, \frac{\pi}{2}\right]
Convex at the intervals
(−∞,−π2]∪[π2,∞)\left(-\infty, - \frac{\pi}{2}\right] \cup \left[\frac{\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
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(−6cos⁡(x)+cot⁡(x))y = \lim_{x \to -\infty}\left(- 6 \cos{\left(x \right)} + \cot{\left(x \right)}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(−6cos⁡(x)+cot⁡(x))y = \lim_{x \to \infty}\left(- 6 \cos{\left(x \right)} + \cot{\left(x \right)}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(x) - 6*cos(x), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(−6cos⁡(x)+cot⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{- 6 \cos{\left(x \right)} + \cot{\left(x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(−6cos⁡(x)+cot⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{- 6 \cos{\left(x \right)} + \cot{\left(x \right)}}{x}\right)
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:
−6cos⁡(x)+cot⁡(x)=−6cos⁡(x)−cot⁡(x)- 6 \cos{\left(x \right)} + \cot{\left(x \right)} = - 6 \cos{\left(x \right)} - \cot{\left(x \right)}
- No
−6cos⁡(x)+cot⁡(x)=6cos⁡(x)+cot⁡(x)- 6 \cos{\left(x \right)} + \cot{\left(x \right)} = 6 \cos{\left(x \right)} + \cot{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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