Mister Exam
Graphing y = ctg(3*x-(4*pi)/3)
The solution
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/ 4*pi\
f(x) = cot|3*x - ----|
\ 3 /
$$f{\left(x \right)} = \cot{\left(3 x - \frac{4 \pi}{3} \right)}$$
f = cot(3*x - 4*pi/3)
The graph of the function
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:
$$\cot{\left(3 x - \frac{4 \pi}{3} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{18}$$
Numerical solution
$$x_{1} = 63.704517697793$$
$$x_{2} = 82.5540736193318$$
$$x_{3} = -44.1568300754565$$
$$x_{4} = 21.8166156499291$$
$$x_{5} = -68.2423737529783$$
$$x_{6} = -35.7792496658838$$
$$x_{7} = 91.9788515801012$$
$$x_{8} = -17.9768912955416$$
$$x_{9} = 19.7222205475359$$
$$x_{10} = 98.2620368872808$$
$$x_{11} = 74.176493209759$$
$$x_{12} = 96.1676417848876$$
$$x_{13} = 85.6956662729216$$
$$x_{14} = 18.6750229963393$$
$$x_{15} = -95.4695100840898$$
$$x_{16} = 32.2885911618951$$
$$x_{17} = -31.5904594610974$$
$$x_{18} = 45.9021593274509$$
$$x_{19} = -39.9680398706701$$
$$x_{20} = 47.9965544298441$$
$$x_{21} = 41.7133691226645$$
$$x_{22} = -7.50491578357562$$
$$x_{23} = -64.0535835481919$$
$$x_{24} = 28.0998009571087$$
$$x_{25} = 12.3918376891597$$
$$x_{26} = -49.3928178314395$$
$$x_{27} = -5.41052068118242$$
$$x_{28} = 30.1941960595019$$
$$x_{29} = -93.3751149816966$$
$$x_{30} = 54.2797397370236$$
$$x_{31} = 8.20304748437335$$
$$x_{32} = 56.3741348394168$$
$$x_{33} = 80.4596785169386$$
$$x_{34} = -97.563905186483$$
$$x_{35} = -4.36332312998582$$
$$x_{36} = -26.3544717051144$$
$$x_{37} = -22.165681500328$$
$$x_{38} = -86.0447321233204$$
$$x_{39} = -66.1479786505851$$
$$x_{40} = -92.3279174305$$
$$x_{41} = -46.2512251778497$$
$$x_{42} = 69.9877030049726$$
$$x_{43} = 16.5806278939461$$
$$x_{44} = 58.46852994181$$
$$x_{45} = 60.5629250442032$$
$$x_{46} = 36.4773813666815$$
$$x_{47} = 1.91986217719376$$
$$x_{48} = 14.4862327915529$$
$$x_{49} = 87.7900613753148$$
$$x_{50} = -79.7615468161409$$
$$x_{51} = -37.873644768277$$
$$x_{52} = 26.0054058547155$$
$$x_{53} = -42.0624349730633$$
$$x_{54} = 52.1853446346305$$
$$x_{55} = -13.7881010907552$$
$$x_{56} = 43.8077642250577$$
$$x_{57} = -73.4783615089613$$
$$x_{58} = -20.0712863979348$$
$$x_{59} = 65.7989128001862$$
$$x_{60} = 10.2974425867665$$
$$x_{61} = 84.648468721725$$
$$x_{62} = -99.6583002888762$$
$$x_{63} = -77.6671517137477$$
$$x_{64} = -0.174532925199433$$
$$x_{65} = -59.8647933434055$$
$$x_{66} = -71.3839664065681$$
$$x_{67} = 78.3652834145454$$
$$x_{68} = 50.0909495322373$$
$$x_{69} = 94.0732466824944$$
$$x_{70} = -29.4960643587042$$
$$x_{71} = -11.693705988362$$
$$x_{72} = -48.3456202802429$$
$$x_{73} = -61.9591884457987$$
$$x_{74} = 76.2708883121522$$
$$x_{75} = -51.4872129338327$$
$$x_{76} = -24.2600766027212$$
$$x_{77} = 89.884456477708$$
$$x_{78} = -81.8559419185341$$
$$x_{79} = 38.5717764690747$$
$$x_{80} = -57.7703982410123$$
$$x_{81} = 40.6661715714679$$
$$x_{82} = 62.6573201465964$$
$$x_{83} = 4.01425727958696$$
$$x_{84} = 100.356431989674$$
$$x_{85} = 6.10865238198015$$
$$x_{86} = 23.9110107523223$$
$$x_{87} = -9.59931088596881$$
$$x_{88} = -90.2335223281068$$
$$x_{89} = -75.5727566113545$$
$$x_{90} = -83.9503370209273$$
$$x_{91} = -27.401669256311$$
$$x_{92} = -88.1391272257137$$
$$x_{93} = 72.0820981073658$$
$$x_{94} = -2.26892802759263$$
$$x_{95} = -15.8824961931484$$
$$x_{96} = -55.6760031386191$$
$$x_{97} = -53.5816080362259$$
$$x_{98} = -33.6848545634906$$
$$x_{99} = -70.3367688553715$$
$$x_{100} = 67.8933079025794$$
$$x_{101} = 34.3829862642883$$
so we need to solve the equation:
$$\cot{\left(3 x - \frac{4 \pi}{3} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{18}$$
Numerical solution
$$x_{1} = 63.704517697793$$
$$x_{2} = 82.5540736193318$$
$$x_{3} = -44.1568300754565$$
$$x_{4} = 21.8166156499291$$
$$x_{5} = -68.2423737529783$$
$$x_{6} = -35.7792496658838$$
$$x_{7} = 91.9788515801012$$
$$x_{8} = -17.9768912955416$$
$$x_{9} = 19.7222205475359$$
$$x_{10} = 98.2620368872808$$
$$x_{11} = 74.176493209759$$
$$x_{12} = 96.1676417848876$$
$$x_{13} = 85.6956662729216$$
$$x_{14} = 18.6750229963393$$
$$x_{15} = -95.4695100840898$$
$$x_{16} = 32.2885911618951$$
$$x_{17} = -31.5904594610974$$
$$x_{18} = 45.9021593274509$$
$$x_{19} = -39.9680398706701$$
$$x_{20} = 47.9965544298441$$
$$x_{21} = 41.7133691226645$$
$$x_{22} = -7.50491578357562$$
$$x_{23} = -64.0535835481919$$
$$x_{24} = 28.0998009571087$$
$$x_{25} = 12.3918376891597$$
$$x_{26} = -49.3928178314395$$
$$x_{27} = -5.41052068118242$$
$$x_{28} = 30.1941960595019$$
$$x_{29} = -93.3751149816966$$
$$x_{30} = 54.2797397370236$$
$$x_{31} = 8.20304748437335$$
$$x_{32} = 56.3741348394168$$
$$x_{33} = 80.4596785169386$$
$$x_{34} = -97.563905186483$$
$$x_{35} = -4.36332312998582$$
$$x_{36} = -26.3544717051144$$
$$x_{37} = -22.165681500328$$
$$x_{38} = -86.0447321233204$$
$$x_{39} = -66.1479786505851$$
$$x_{40} = -92.3279174305$$
$$x_{41} = -46.2512251778497$$
$$x_{42} = 69.9877030049726$$
$$x_{43} = 16.5806278939461$$
$$x_{44} = 58.46852994181$$
$$x_{45} = 60.5629250442032$$
$$x_{46} = 36.4773813666815$$
$$x_{47} = 1.91986217719376$$
$$x_{48} = 14.4862327915529$$
$$x_{49} = 87.7900613753148$$
$$x_{50} = -79.7615468161409$$
$$x_{51} = -37.873644768277$$
$$x_{52} = 26.0054058547155$$
$$x_{53} = -42.0624349730633$$
$$x_{54} = 52.1853446346305$$
$$x_{55} = -13.7881010907552$$
$$x_{56} = 43.8077642250577$$
$$x_{57} = -73.4783615089613$$
$$x_{58} = -20.0712863979348$$
$$x_{59} = 65.7989128001862$$
$$x_{60} = 10.2974425867665$$
$$x_{61} = 84.648468721725$$
$$x_{62} = -99.6583002888762$$
$$x_{63} = -77.6671517137477$$
$$x_{64} = -0.174532925199433$$
$$x_{65} = -59.8647933434055$$
$$x_{66} = -71.3839664065681$$
$$x_{67} = 78.3652834145454$$
$$x_{68} = 50.0909495322373$$
$$x_{69} = 94.0732466824944$$
$$x_{70} = -29.4960643587042$$
$$x_{71} = -11.693705988362$$
$$x_{72} = -48.3456202802429$$
$$x_{73} = -61.9591884457987$$
$$x_{74} = 76.2708883121522$$
$$x_{75} = -51.4872129338327$$
$$x_{76} = -24.2600766027212$$
$$x_{77} = 89.884456477708$$
$$x_{78} = -81.8559419185341$$
$$x_{79} = 38.5717764690747$$
$$x_{80} = -57.7703982410123$$
$$x_{81} = 40.6661715714679$$
$$x_{82} = 62.6573201465964$$
$$x_{83} = 4.01425727958696$$
$$x_{84} = 100.356431989674$$
$$x_{85} = 6.10865238198015$$
$$x_{86} = 23.9110107523223$$
$$x_{87} = -9.59931088596881$$
$$x_{88} = -90.2335223281068$$
$$x_{89} = -75.5727566113545$$
$$x_{90} = -83.9503370209273$$
$$x_{91} = -27.401669256311$$
$$x_{92} = -88.1391272257137$$
$$x_{93} = 72.0820981073658$$
$$x_{94} = -2.26892802759263$$
$$x_{95} = -15.8824961931484$$
$$x_{96} = -55.6760031386191$$
$$x_{97} = -53.5816080362259$$
$$x_{98} = -33.6848545634906$$
$$x_{99} = -70.3367688553715$$
$$x_{100} = 67.8933079025794$$
$$x_{101} = 34.3829862642883$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- 3 \cot^{2}{\left(3 x - \frac{4 \pi}{3} \right)} - 3 = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- 3 \cot^{2}{\left(3 x - \frac{4 \pi}{3} \right)} - 3 = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$- 18 \left(\tan^{2}{\left(3 x + \frac{\pi}{6} \right)} + 1\right) \tan{\left(3 x + \frac{\pi}{6} \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{18}$$
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
$$\left(-\infty, - \frac{\pi}{18}\right]$$
Convex at the intervals
$$\left[- \frac{\pi}{18}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$- 18 \left(\tan^{2}{\left(3 x + \frac{\pi}{6} \right)} + 1\right) \tan{\left(3 x + \frac{\pi}{6} \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{18}$$
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
$$\left(-\infty, - \frac{\pi}{18}\right]$$
Convex at the intervals
$$\left[- \frac{\pi}{18}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \cot{\left(3 x - \frac{4 \pi}{3} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \cot{\left(3 x - \frac{4 \pi}{3} \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \cot{\left(3 x - \frac{4 \pi}{3} \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \cot{\left(3 x - \frac{4 \pi}{3} \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(3*x - 4*pi/3), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(3 x - \frac{4 \pi}{3} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(3 x - \frac{4 \pi}{3} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(3 x - \frac{4 \pi}{3} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(3 x - \frac{4 \pi}{3} \right)}}{x}\right)$$
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:
$$\cot{\left(3 x - \frac{4 \pi}{3} \right)} = - \cot{\left(3 x + \frac{4 \pi}{3} \right)}$$
- No
$$\cot{\left(3 x - \frac{4 \pi}{3} \right)} = \cot{\left(3 x + \frac{4 \pi}{3} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\cot{\left(3 x - \frac{4 \pi}{3} \right)} = - \cot{\left(3 x + \frac{4 \pi}{3} \right)}$$
- No
$$\cot{\left(3 x - \frac{4 \pi}{3} \right)} = \cot{\left(3 x + \frac{4 \pi}{3} \right)}$$
- No
so, the function
not is
neither even, nor odd