Mister Exam
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- Graph of implicit function
- Derivative Step by Step
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- Graph of parametric function
- Graph in Polar Coordinates
- Surface defined parametrically
- Surface defined by equation
- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- x^4-2x^2-3
- x^3-2x^2+x
- (x^2-4)/x
-
x^2-4x
-
Identical expressions
- (sin5xcos3x-sin3xcos5x)/|cosx|
- ( sinus of 5x co sinus of e of 3x minus sinus of 3x co sinus of e of 5x) divide by module of co sinus of e of x|
- sin5xcos3x-sin3xcos5x/|cosx|
- (sin5xcos3x-sin3xcos5x) divide by |cosx|
-
Similar expressions
- (sin5xcos3x+sin3xcos5x)/|cosx|
Graphing y = (sin5xcos3x-sin3xcos5x)/|cosx|
The solution
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[src]
sin(5*x)*cos(3*x) - sin(3*x)*cos(5*x)
f(x) = -------------------------------------
|cos(x)|
$$f{\left(x \right)} = \frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|}$$
f = (-sin(3*x)*cos(5*x) + sin(5*x)*cos(3*x))/Abs(cos(x))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 1.5707963267949$$
$$x_{2} = 4.71238898038469$$
$$x_{1} = 1.5707963267949$$
$$x_{2} = 4.71238898038469$$
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:
$$\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 43.9822971502571$$
$$x_{2} = -97.3893722612836$$
$$x_{3} = -43.9822971502571$$
$$x_{4} = -72.2566310325652$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = 81.6814089933346$$
$$x_{7} = -31.4159265358979$$
$$x_{8} = -433.539786195391$$
$$x_{9} = -78.5398163397448$$
$$x_{10} = 97.3893722612836$$
$$x_{11} = 9.42477796076938$$
$$x_{12} = 44921.6333536805$$
$$x_{13} = -25.1327412287183$$
$$x_{14} = -295.309709437441$$
$$x_{15} = -21.9911485751286$$
$$x_{16} = -94.2477796076938$$
$$x_{17} = 6.28318530717959$$
$$x_{18} = 3.14159265358979$$
$$x_{19} = -50.2654824574367$$
$$x_{20} = 28.2743338823081$$
$$x_{21} = -75.398223686155$$
$$x_{22} = -28.2743338823081$$
$$x_{23} = -56.5486677646163$$
$$x_{24} = -65.9734457253857$$
$$x_{25} = -40.8407044966673$$
$$x_{26} = -91.106186954104$$
$$x_{27} = 50.2654824574367$$
$$x_{28} = -116.238928182822$$
$$x_{29} = -69.1150383789755$$
$$x_{30} = -100.530964914873$$
$$x_{31} = 56.5486677646163$$
$$x_{32} = -62.8318530717959$$
$$x_{33} = 40.8407044966673$$
$$x_{34} = 100.530964914873$$
$$x_{35} = 18.8495559215388$$
$$x_{36} = 62.8318530717959$$
$$x_{37} = -53.4070751110265$$
$$x_{38} = 94.2477796076938$$
$$x_{39} = -3.14159265358979$$
$$x_{40} = 12.5663706143592$$
$$x_{41} = -84.8230016469244$$
$$x_{42} = 34.5575191894877$$
$$x_{43} = 47.1238898038469$$
$$x_{44} = -15.707963267949$$
$$x_{45} = 53.4070751110265$$
$$x_{46} = 65.9734457253857$$
$$x_{47} = 87.9645943005142$$
$$x_{48} = 91.106186954104$$
$$x_{49} = 59.6902604182061$$
$$x_{50} = 69.1150383789755$$
$$x_{51} = -6.28318530717959$$
$$x_{52} = 75.398223686155$$
$$x_{53} = -37.6991118430775$$
$$x_{54} = -12.5663706143592$$
$$x_{55} = -18.8495559215388$$
$$x_{56} = 31.4159265358979$$
$$x_{57} = -81.6814089933346$$
$$x_{58} = 78.5398163397448$$
$$x_{59} = -182.212373908208$$
$$x_{60} = 15.707963267949$$
$$x_{61} = 37.6991118430775$$
$$x_{62} = 25.1327412287183$$
$$x_{63} = -47.1238898038469$$
$$x_{64} = -358.141562509236$$
$$x_{65} = 0$$
$$x_{66} = -9.42477796076938$$
$$x_{67} = 223.053078404875$$
so we need to solve the equation:
$$\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 43.9822971502571$$
$$x_{2} = -97.3893722612836$$
$$x_{3} = -43.9822971502571$$
$$x_{4} = -72.2566310325652$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = 81.6814089933346$$
$$x_{7} = -31.4159265358979$$
$$x_{8} = -433.539786195391$$
$$x_{9} = -78.5398163397448$$
$$x_{10} = 97.3893722612836$$
$$x_{11} = 9.42477796076938$$
$$x_{12} = 44921.6333536805$$
$$x_{13} = -25.1327412287183$$
$$x_{14} = -295.309709437441$$
$$x_{15} = -21.9911485751286$$
$$x_{16} = -94.2477796076938$$
$$x_{17} = 6.28318530717959$$
$$x_{18} = 3.14159265358979$$
$$x_{19} = -50.2654824574367$$
$$x_{20} = 28.2743338823081$$
$$x_{21} = -75.398223686155$$
$$x_{22} = -28.2743338823081$$
$$x_{23} = -56.5486677646163$$
$$x_{24} = -65.9734457253857$$
$$x_{25} = -40.8407044966673$$
$$x_{26} = -91.106186954104$$
$$x_{27} = 50.2654824574367$$
$$x_{28} = -116.238928182822$$
$$x_{29} = -69.1150383789755$$
$$x_{30} = -100.530964914873$$
$$x_{31} = 56.5486677646163$$
$$x_{32} = -62.8318530717959$$
$$x_{33} = 40.8407044966673$$
$$x_{34} = 100.530964914873$$
$$x_{35} = 18.8495559215388$$
$$x_{36} = 62.8318530717959$$
$$x_{37} = -53.4070751110265$$
$$x_{38} = 94.2477796076938$$
$$x_{39} = -3.14159265358979$$
$$x_{40} = 12.5663706143592$$
$$x_{41} = -84.8230016469244$$
$$x_{42} = 34.5575191894877$$
$$x_{43} = 47.1238898038469$$
$$x_{44} = -15.707963267949$$
$$x_{45} = 53.4070751110265$$
$$x_{46} = 65.9734457253857$$
$$x_{47} = 87.9645943005142$$
$$x_{48} = 91.106186954104$$
$$x_{49} = 59.6902604182061$$
$$x_{50} = 69.1150383789755$$
$$x_{51} = -6.28318530717959$$
$$x_{52} = 75.398223686155$$
$$x_{53} = -37.6991118430775$$
$$x_{54} = -12.5663706143592$$
$$x_{55} = -18.8495559215388$$
$$x_{56} = 31.4159265358979$$
$$x_{57} = -81.6814089933346$$
$$x_{58} = 78.5398163397448$$
$$x_{59} = -182.212373908208$$
$$x_{60} = 15.707963267949$$
$$x_{61} = 37.6991118430775$$
$$x_{62} = 25.1327412287183$$
$$x_{63} = -47.1238898038469$$
$$x_{64} = -358.141562509236$$
$$x_{65} = 0$$
$$x_{66} = -9.42477796076938$$
$$x_{67} = 223.053078404875$$
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
$$\frac{2 \sin{\left(3 x \right)} \sin{\left(5 x \right)} + 2 \cos{\left(3 x \right)} \cos{\left(5 x \right)}}{\left|{\cos{\left(x \right)}}\right|} + \frac{\left(- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} = 0$$
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
$$\frac{2 \sin{\left(3 x \right)} \sin{\left(5 x \right)} + 2 \cos{\left(3 x \right)} \cos{\left(5 x \right)}}{\left|{\cos{\left(x \right)}}\right|} + \frac{\left(- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} = 0$$
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
$$\frac{4 \left(\sin{\left(3 x \right)} \sin{\left(5 x \right)} + \cos{\left(3 x \right)} \cos{\left(5 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \left(- \frac{2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right)}{\cos{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \operatorname{sign}{\left(\cos{\left(x \right)} \right)}\right)}{\cos{\left(x \right)}} + \frac{4 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)}{\left|{\cos{\left(x \right)}}\right|} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
$$x_{2} = \pi$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1.5707963267949$$
$$x_{2} = 4.71238898038469$$
$$\lim_{x \to 1.5707963267949^-}\left(\frac{4 \left(\sin{\left(3 x \right)} \sin{\left(5 x \right)} + \cos{\left(3 x \right)} \cos{\left(5 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \left(- \frac{2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right)}{\cos{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \operatorname{sign}{\left(\cos{\left(x \right)} \right)}\right)}{\cos{\left(x \right)}} + \frac{4 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)}{\left|{\cos{\left(x \right)}}\right|}\right) = 2.88230376151712 \cdot 10^{17}$$
$$\lim_{x \to 1.5707963267949^+}\left(\frac{4 \left(\sin{\left(3 x \right)} \sin{\left(5 x \right)} + \cos{\left(3 x \right)} \cos{\left(5 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \left(- \frac{2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right)}{\cos{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \operatorname{sign}{\left(\cos{\left(x \right)} \right)}\right)}{\cos{\left(x \right)}} + \frac{4 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)}{\left|{\cos{\left(x \right)}}\right|}\right) = 2.88230376151712 \cdot 10^{17}$$
- limits are equal, then skip the corresponding point
$$\lim_{x \to 4.71238898038469^-}\left(\frac{4 \left(\sin{\left(3 x \right)} \sin{\left(5 x \right)} + \cos{\left(3 x \right)} \cos{\left(5 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \left(- \frac{2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right)}{\cos{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \operatorname{sign}{\left(\cos{\left(x \right)} \right)}\right)}{\cos{\left(x \right)}} + \frac{4 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)}{\left|{\cos{\left(x \right)}}\right|}\right) = 5.73130968608109 \cdot 10^{32}$$
$$\lim_{x \to 4.71238898038469^+}\left(\frac{4 \left(\sin{\left(3 x \right)} \sin{\left(5 x \right)} + \cos{\left(3 x \right)} \cos{\left(5 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \left(- \frac{2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right)}{\cos{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \operatorname{sign}{\left(\cos{\left(x \right)} \right)}\right)}{\cos{\left(x \right)}} + \frac{4 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)}{\left|{\cos{\left(x \right)}}\right|}\right) = 5.73130968608109 \cdot 10^{32}$$
- limits are equal, then skip the corresponding point
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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[\pi, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{4 \left(\sin{\left(3 x \right)} \sin{\left(5 x \right)} + \cos{\left(3 x \right)} \cos{\left(5 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \left(- \frac{2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right)}{\cos{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \operatorname{sign}{\left(\cos{\left(x \right)} \right)}\right)}{\cos{\left(x \right)}} + \frac{4 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)}{\left|{\cos{\left(x \right)}}\right|} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
$$x_{2} = \pi$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1.5707963267949$$
$$x_{2} = 4.71238898038469$$
$$\lim_{x \to 1.5707963267949^-}\left(\frac{4 \left(\sin{\left(3 x \right)} \sin{\left(5 x \right)} + \cos{\left(3 x \right)} \cos{\left(5 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \left(- \frac{2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right)}{\cos{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \operatorname{sign}{\left(\cos{\left(x \right)} \right)}\right)}{\cos{\left(x \right)}} + \frac{4 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)}{\left|{\cos{\left(x \right)}}\right|}\right) = 2.88230376151712 \cdot 10^{17}$$
$$\lim_{x \to 1.5707963267949^+}\left(\frac{4 \left(\sin{\left(3 x \right)} \sin{\left(5 x \right)} + \cos{\left(3 x \right)} \cos{\left(5 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \left(- \frac{2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right)}{\cos{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \operatorname{sign}{\left(\cos{\left(x \right)} \right)}\right)}{\cos{\left(x \right)}} + \frac{4 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)}{\left|{\cos{\left(x \right)}}\right|}\right) = 2.88230376151712 \cdot 10^{17}$$
- limits are equal, then skip the corresponding point
$$\lim_{x \to 4.71238898038469^-}\left(\frac{4 \left(\sin{\left(3 x \right)} \sin{\left(5 x \right)} + \cos{\left(3 x \right)} \cos{\left(5 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \left(- \frac{2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right)}{\cos{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \operatorname{sign}{\left(\cos{\left(x \right)} \right)}\right)}{\cos{\left(x \right)}} + \frac{4 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)}{\left|{\cos{\left(x \right)}}\right|}\right) = 5.73130968608109 \cdot 10^{32}$$
$$\lim_{x \to 4.71238898038469^+}\left(\frac{4 \left(\sin{\left(3 x \right)} \sin{\left(5 x \right)} + \cos{\left(3 x \right)} \cos{\left(5 x \right)}\right) \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right) \left(- \frac{2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right)}{\cos{\left(x \right)}} + \frac{2 \sin^{2}{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \operatorname{sign}{\left(\cos{\left(x \right)} \right)}\right)}{\cos{\left(x \right)}} + \frac{4 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)}{\left|{\cos{\left(x \right)}}\right|}\right) = 5.73130968608109 \cdot 10^{32}$$
- limits are equal, then skip the corresponding point
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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[\pi, \infty\right)$$
Vertical asymptotes
Have:
$$x_{1} = 1.5707963267949$$
$$x_{2} = 4.71238898038469$$
$$x_{1} = 1.5707963267949$$
$$x_{2} = 4.71238898038469$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|}\right) = \frac{\left\langle -2, 2\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\left\langle -2, 2\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}$$
$$\lim_{x \to \infty}\left(\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|}\right) = \frac{\left\langle -2, 2\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\left\langle -2, 2\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}$$
$$\lim_{x \to -\infty}\left(\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|}\right) = \frac{\left\langle -2, 2\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \frac{\left\langle -2, 2\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}$$
$$\lim_{x \to \infty}\left(\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|}\right) = \frac{\left\langle -2, 2\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \frac{\left\langle -2, 2\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (sin(5*x)*cos(3*x) - sin(3*x)*cos(5*x))/Abs(cos(x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{x \left|{\cos{\left(x \right)}}\right|}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{x \left|{\cos{\left(x \right)}}\right|}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{x \left|{\cos{\left(x \right)}}\right|}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{x \left|{\cos{\left(x \right)}}\right|}\right) = 0$$
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:
$$\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|} = \frac{\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|}$$
- No
$$\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|} = - \frac{\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|} = \frac{\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|}$$
- No
$$\frac{- \sin{\left(3 x \right)} \cos{\left(5 x \right)} + \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|} = - \frac{\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}}{\left|{\cos{\left(x \right)}}\right|}$$
- No
so, the function
not is
neither even, nor odd