Mister Exam
Graphing y = -1,09*cos((pi*x)/(1,5))+0,197*cos((pi*x)/2*1,5)^2
The solution
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/pi*x \
|----*3|
/pi*x\ 2| 2 |
109*cos|----| 197*cos |------|
\3/2 / \ 2 /
f(x) = - ------------- + ----------------
100 1000
$$f{\left(x \right)} = \frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}$$
f = 197*cos(3*((pi*x)/2)/2)^2/1000 - 109*cos((pi*x)/(3/2))/100
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:
$$\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 92.2999900213483$$
$$x_{2} = -15.7000099786517$$
$$x_{3} = -99.7000099786517$$
$$x_{4} = 14.2830436872061$$
$$x_{5} = 56.2999900213483$$
$$x_{6} = -50.2830436872061$$
$$x_{7} = -14.2830436872061$$
$$x_{8} = -45.7169563127939$$
$$x_{9} = 99.7000099786517$$
$$x_{10} = -93.7169563127939$$
$$x_{11} = -69.7169563127939$$
$$x_{12} = -98.2830436872061$$
$$x_{13} = -9.71695631279395$$
$$x_{14} = 54.6632347956599$$
$$x_{15} = -62.2830436872061$$
$$x_{16} = 26.2830436872061$$
$$x_{17} = -51.7000099786517$$
$$x_{18} = 62.2830436872061$$
$$x_{19} = -32.2999900213483$$
$$x_{20} = 20.2999900213483$$
$$x_{21} = -26.2830436872061$$
$$x_{22} = 32.2999900213483$$
$$x_{23} = -21.7169563127939$$
$$x_{24} = -20.2999900213483$$
$$x_{25} = 68.2999900213483$$
$$x_{26} = 66.6632347956599$$
$$x_{27} = 86.2830436872061$$
$$x_{28} = -2.28304368720605$$
$$x_{29} = -29.3367652043401$$
$$x_{30} = -63.7000099786517$$
$$x_{31} = -74.2830436872061$$
$$x_{32} = -3.70000997865169$$
$$x_{33} = 21.7169563127939$$
$$x_{34} = 57.7169563127939$$
$$x_{35} = 2.28304368720605$$
$$x_{36} = 63.7000099786517$$
$$x_{37} = 80.2999900213483$$
$$x_{38} = -86.2830436872061$$
$$x_{39} = 50.2830436872061$$
$$x_{40} = -75.7000099786517$$
$$x_{41} = -92.2999900213483$$
$$x_{42} = -77.3367652043401$$
$$x_{43} = 33.7169563127939$$
$$x_{44} = 87.7000099786517$$
$$x_{45} = 51.7000099786517$$
$$x_{46} = 9.71695631279395$$
$$x_{47} = -53.3367652043401$$
$$x_{48} = 6.66323479565987$$
$$x_{49} = 90.6632347956599$$
$$x_{50} = 74.2830436872061$$
$$x_{51} = 78.6632347956599$$
$$x_{52} = -57.7169563127939$$
$$x_{53} = -41.3367652043401$$
$$x_{54} = 102.66323479566$$
$$x_{55} = -89.3367652043401$$
$$x_{56} = -5.33676520434013$$
$$x_{57} = -81.7169563127939$$
$$x_{58} = -38.2830436872061$$
$$x_{59} = -87.7000099786517$$
$$x_{60} = -56.2999900213483$$
$$x_{61} = -65.3367652043401$$
$$x_{62} = 45.7169563127939$$
$$x_{63} = 39.7000099786517$$
$$x_{64} = 3.70000997865169$$
$$x_{65} = -80.2999900213483$$
$$x_{66} = 98.2830436872061$$
$$x_{67} = 18.6632347956599$$
$$x_{68} = 75.7000099786517$$
$$x_{69} = -17.3367652043401$$
$$x_{70} = 42.6632347956599$$
$$x_{71} = 93.7169563127939$$
$$x_{72} = 44.2999900213483$$
$$x_{73} = -39.7000099786517$$
$$x_{74} = 15.7000099786517$$
$$x_{75} = -33.7169563127939$$
$$x_{76} = 81.7169563127939$$
$$x_{77} = -8.29999002134831$$
$$x_{78} = 30.6632347956599$$
$$x_{79} = 27.7000099786517$$
$$x_{80} = 38.2830436872061$$
$$x_{81} = -68.2999900213483$$
$$x_{82} = 69.7169563127939$$
$$x_{83} = 8.29999002134831$$
$$x_{84} = -27.7000099786517$$
$$x_{85} = -44.2999900213483$$
so we need to solve the equation:
$$\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 92.2999900213483$$
$$x_{2} = -15.7000099786517$$
$$x_{3} = -99.7000099786517$$
$$x_{4} = 14.2830436872061$$
$$x_{5} = 56.2999900213483$$
$$x_{6} = -50.2830436872061$$
$$x_{7} = -14.2830436872061$$
$$x_{8} = -45.7169563127939$$
$$x_{9} = 99.7000099786517$$
$$x_{10} = -93.7169563127939$$
$$x_{11} = -69.7169563127939$$
$$x_{12} = -98.2830436872061$$
$$x_{13} = -9.71695631279395$$
$$x_{14} = 54.6632347956599$$
$$x_{15} = -62.2830436872061$$
$$x_{16} = 26.2830436872061$$
$$x_{17} = -51.7000099786517$$
$$x_{18} = 62.2830436872061$$
$$x_{19} = -32.2999900213483$$
$$x_{20} = 20.2999900213483$$
$$x_{21} = -26.2830436872061$$
$$x_{22} = 32.2999900213483$$
$$x_{23} = -21.7169563127939$$
$$x_{24} = -20.2999900213483$$
$$x_{25} = 68.2999900213483$$
$$x_{26} = 66.6632347956599$$
$$x_{27} = 86.2830436872061$$
$$x_{28} = -2.28304368720605$$
$$x_{29} = -29.3367652043401$$
$$x_{30} = -63.7000099786517$$
$$x_{31} = -74.2830436872061$$
$$x_{32} = -3.70000997865169$$
$$x_{33} = 21.7169563127939$$
$$x_{34} = 57.7169563127939$$
$$x_{35} = 2.28304368720605$$
$$x_{36} = 63.7000099786517$$
$$x_{37} = 80.2999900213483$$
$$x_{38} = -86.2830436872061$$
$$x_{39} = 50.2830436872061$$
$$x_{40} = -75.7000099786517$$
$$x_{41} = -92.2999900213483$$
$$x_{42} = -77.3367652043401$$
$$x_{43} = 33.7169563127939$$
$$x_{44} = 87.7000099786517$$
$$x_{45} = 51.7000099786517$$
$$x_{46} = 9.71695631279395$$
$$x_{47} = -53.3367652043401$$
$$x_{48} = 6.66323479565987$$
$$x_{49} = 90.6632347956599$$
$$x_{50} = 74.2830436872061$$
$$x_{51} = 78.6632347956599$$
$$x_{52} = -57.7169563127939$$
$$x_{53} = -41.3367652043401$$
$$x_{54} = 102.66323479566$$
$$x_{55} = -89.3367652043401$$
$$x_{56} = -5.33676520434013$$
$$x_{57} = -81.7169563127939$$
$$x_{58} = -38.2830436872061$$
$$x_{59} = -87.7000099786517$$
$$x_{60} = -56.2999900213483$$
$$x_{61} = -65.3367652043401$$
$$x_{62} = 45.7169563127939$$
$$x_{63} = 39.7000099786517$$
$$x_{64} = 3.70000997865169$$
$$x_{65} = -80.2999900213483$$
$$x_{66} = 98.2830436872061$$
$$x_{67} = 18.6632347956599$$
$$x_{68} = 75.7000099786517$$
$$x_{69} = -17.3367652043401$$
$$x_{70} = 42.6632347956599$$
$$x_{71} = 93.7169563127939$$
$$x_{72} = 44.2999900213483$$
$$x_{73} = -39.7000099786517$$
$$x_{74} = 15.7000099786517$$
$$x_{75} = -33.7169563127939$$
$$x_{76} = 81.7169563127939$$
$$x_{77} = -8.29999002134831$$
$$x_{78} = 30.6632347956599$$
$$x_{79} = 27.7000099786517$$
$$x_{80} = 38.2830436872061$$
$$x_{81} = -68.2999900213483$$
$$x_{82} = 69.7169563127939$$
$$x_{83} = 8.29999002134831$$
$$x_{84} = -27.7000099786517$$
$$x_{85} = -44.2999900213483$$
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
$$\frac{\pi^{2} \left(15957 \sin^{2}{\left(\frac{3 \pi x}{4} \right)} + 34880 \cos{\left(\frac{2 \pi x}{3} \right)} - 15957 \cos^{2}{\left(\frac{3 \pi x}{4} \right)}\right)}{72000} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -72.8737790478197$$
$$x_{2} = 30.5555182502932$$
$$x_{3} = 78.5555182502932$$
$$x_{4} = -312.87377904782$$
$$x_{5} = -357.975199088872$$
$$x_{6} = -53.4444817497068$$
$$x_{7} = -69.9751990888724$$
$$x_{8} = 33.9751990888724$$
$$x_{9} = -236.292165263802$$
$$x_{10} = -15.7078347361982$$
$$x_{11} = -17.4444817497068$$
$$x_{12} = 80.2921652638018$$
$$x_{13} = -389.444481749707$$
$$x_{14} = 20.2921652638018$$
$$x_{15} = 44.2921652638018$$
$$x_{16} = -93.9751990888724$$
$$x_{17} = -38.0248009111276$$
$$x_{18} = -62.0248009111276$$
$$x_{19} = -96.8737790478197$$
$$x_{20} = 59.1262209521803$$
$$x_{21} = -29.4444817497068$$
$$x_{22} = 45.9751990888724$$
$$x_{23} = -51.7078347361982$$
$$x_{24} = 57.9751990888724$$
$$x_{25} = -24.8737790478197$$
$$x_{26} = 11.1262209521803$$
$$x_{27} = -98.0248009111276$$
$$x_{28} = -74.0248009111276$$
$$x_{29} = -26.0248009111276$$
$$x_{30} = -279.707834736198$$
$$x_{31} = -14.0248009111276$$
$$x_{32} = -9.97519908887245$$
$$x_{33} = -81.9751990888724$$
$$x_{34} = -86.0248009111276$$
$$x_{35} = 92.2921652638018$$
$$x_{36} = -99.7078347361982$$
$$x_{37} = -41.4444817497068$$
$$x_{38} = -77.4444817497068$$
$$x_{39} = 98.0248009111276$$
$$x_{40} = 47.1262209521803$$
$$x_{41} = 2.02480091112755$$
$$x_{42} = 35.1262209521803$$
$$x_{43} = -60.8737790478197$$
$$x_{44} = -0.873779047819721$$
$$x_{45} = 66.5555182502932$$
$$x_{46} = 42.5555182502932$$
$$x_{47} = 54.5555182502932$$
$$x_{48} = 50.0248009111276$$
$$x_{49} = -57.9751990888724$$
$$x_{50} = -2.02480091112755$$
$$x_{51} = -84.8737790478197$$
$$x_{52} = -45.9751990888724$$
$$x_{53} = -50.0248009111276$$
$$x_{54} = -89.4444817497068$$
$$x_{55} = 23.1262209521803$$
$$x_{56} = 86.0248009111276$$
$$x_{57} = -3.70783473619817$$
$$x_{58} = 18.5555182502932$$
$$x_{59} = 9.97519908887245$$
$$x_{60} = -27.7078347361982$$
$$x_{61} = 95.1262209521803$$
$$x_{62} = -5.44448174970676$$
$$x_{63} = 26.0248009111276$$
$$x_{64} = -12.8737790478197$$
$$x_{65} = -33.9751990888724$$
$$x_{66} = -65.4444817497068$$
$$x_{67} = 90.5555182502932$$
$$x_{68} = 32.2921652638018$$
$$x_{69} = 81.9751990888724$$
$$x_{70} = -75.7078347361982$$
$$x_{71} = 21.9751990888724$$
$$x_{72} = -368.292165263802$$
$$x_{73} = 6.55551825029324$$
$$x_{74} = 14.0248009111276$$
$$x_{75} = -21.9751990888724$$
$$x_{76} = -329.444481749707$$
$$x_{77} = 83.1262209521803$$
$$x_{78} = 74.0248009111276$$
$$x_{79} = 62.0248009111276$$
$$x_{80} = 71.1262209521803$$
$$x_{81} = -87.7078347361982$$
$$x_{82} = -39.7078347361982$$
$$x_{83} = -36.8737790478197$$
$$x_{84} = 8.29216526380183$$
$$x_{85} = 68.2921652638018$$
$$x_{86} = 93.9751990888724$$
$$x_{87} = -63.7078347361982$$
$$x_{88} = 38.0248009111276$$
$$x_{89} = 56.2921652638018$$
$$x_{90} = -48.8737790478197$$
$$x_{91} = 69.9751990888724$$
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
$$\left[98.0248009111276, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -357.975199088872\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\pi^{2} \left(15957 \sin^{2}{\left(\frac{3 \pi x}{4} \right)} + 34880 \cos{\left(\frac{2 \pi x}{3} \right)} - 15957 \cos^{2}{\left(\frac{3 \pi x}{4} \right)}\right)}{72000} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -72.8737790478197$$
$$x_{2} = 30.5555182502932$$
$$x_{3} = 78.5555182502932$$
$$x_{4} = -312.87377904782$$
$$x_{5} = -357.975199088872$$
$$x_{6} = -53.4444817497068$$
$$x_{7} = -69.9751990888724$$
$$x_{8} = 33.9751990888724$$
$$x_{9} = -236.292165263802$$
$$x_{10} = -15.7078347361982$$
$$x_{11} = -17.4444817497068$$
$$x_{12} = 80.2921652638018$$
$$x_{13} = -389.444481749707$$
$$x_{14} = 20.2921652638018$$
$$x_{15} = 44.2921652638018$$
$$x_{16} = -93.9751990888724$$
$$x_{17} = -38.0248009111276$$
$$x_{18} = -62.0248009111276$$
$$x_{19} = -96.8737790478197$$
$$x_{20} = 59.1262209521803$$
$$x_{21} = -29.4444817497068$$
$$x_{22} = 45.9751990888724$$
$$x_{23} = -51.7078347361982$$
$$x_{24} = 57.9751990888724$$
$$x_{25} = -24.8737790478197$$
$$x_{26} = 11.1262209521803$$
$$x_{27} = -98.0248009111276$$
$$x_{28} = -74.0248009111276$$
$$x_{29} = -26.0248009111276$$
$$x_{30} = -279.707834736198$$
$$x_{31} = -14.0248009111276$$
$$x_{32} = -9.97519908887245$$
$$x_{33} = -81.9751990888724$$
$$x_{34} = -86.0248009111276$$
$$x_{35} = 92.2921652638018$$
$$x_{36} = -99.7078347361982$$
$$x_{37} = -41.4444817497068$$
$$x_{38} = -77.4444817497068$$
$$x_{39} = 98.0248009111276$$
$$x_{40} = 47.1262209521803$$
$$x_{41} = 2.02480091112755$$
$$x_{42} = 35.1262209521803$$
$$x_{43} = -60.8737790478197$$
$$x_{44} = -0.873779047819721$$
$$x_{45} = 66.5555182502932$$
$$x_{46} = 42.5555182502932$$
$$x_{47} = 54.5555182502932$$
$$x_{48} = 50.0248009111276$$
$$x_{49} = -57.9751990888724$$
$$x_{50} = -2.02480091112755$$
$$x_{51} = -84.8737790478197$$
$$x_{52} = -45.9751990888724$$
$$x_{53} = -50.0248009111276$$
$$x_{54} = -89.4444817497068$$
$$x_{55} = 23.1262209521803$$
$$x_{56} = 86.0248009111276$$
$$x_{57} = -3.70783473619817$$
$$x_{58} = 18.5555182502932$$
$$x_{59} = 9.97519908887245$$
$$x_{60} = -27.7078347361982$$
$$x_{61} = 95.1262209521803$$
$$x_{62} = -5.44448174970676$$
$$x_{63} = 26.0248009111276$$
$$x_{64} = -12.8737790478197$$
$$x_{65} = -33.9751990888724$$
$$x_{66} = -65.4444817497068$$
$$x_{67} = 90.5555182502932$$
$$x_{68} = 32.2921652638018$$
$$x_{69} = 81.9751990888724$$
$$x_{70} = -75.7078347361982$$
$$x_{71} = 21.9751990888724$$
$$x_{72} = -368.292165263802$$
$$x_{73} = 6.55551825029324$$
$$x_{74} = 14.0248009111276$$
$$x_{75} = -21.9751990888724$$
$$x_{76} = -329.444481749707$$
$$x_{77} = 83.1262209521803$$
$$x_{78} = 74.0248009111276$$
$$x_{79} = 62.0248009111276$$
$$x_{80} = 71.1262209521803$$
$$x_{81} = -87.7078347361982$$
$$x_{82} = -39.7078347361982$$
$$x_{83} = -36.8737790478197$$
$$x_{84} = 8.29216526380183$$
$$x_{85} = 68.2921652638018$$
$$x_{86} = 93.9751990888724$$
$$x_{87} = -63.7078347361982$$
$$x_{88} = 38.0248009111276$$
$$x_{89} = 56.2921652638018$$
$$x_{90} = -48.8737790478197$$
$$x_{91} = 69.9751990888724$$
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
$$\left[98.0248009111276, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -357.975199088872\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}\right) = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle$$
$$\lim_{x \to \infty}\left(\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}\right) = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle$$
$$\lim_{x \to -\infty}\left(\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}\right) = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle$$
$$\lim_{x \to \infty}\left(\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}\right) = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle - \frac{109}{100}, \frac{1287}{1000}\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -109*cos((pi*x)/(3/2))/100 + 197*cos(((pi*x)/2)*3/2)^2/1000, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100}}{x}\right) = 0$$
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:
$$\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100} = - \frac{109 \cos{\left(\frac{2 \pi x}{3} \right)}}{100} + \frac{197 \cos^{2}{\left(\frac{3 \pi x}{4} \right)}}{1000}$$
- No
$$\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100} = \frac{109 \cos{\left(\frac{2 \pi x}{3} \right)}}{100} - \frac{197 \cos^{2}{\left(\frac{3 \pi x}{4} \right)}}{1000}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100} = - \frac{109 \cos{\left(\frac{2 \pi x}{3} \right)}}{100} + \frac{197 \cos^{2}{\left(\frac{3 \pi x}{4} \right)}}{1000}$$
- No
$$\frac{197 \cos^{2}{\left(\frac{3 \frac{\pi x}{2}}{2} \right)}}{1000} - \frac{109 \cos{\left(\frac{\pi x}{\frac{3}{2}} \right)}}{100} = \frac{109 \cos{\left(\frac{2 \pi x}{3} \right)}}{100} - \frac{197 \cos^{2}{\left(\frac{3 \pi x}{4} \right)}}{1000}$$
- No
so, the function
not is
neither even, nor odd