Mister Exam
Graphing y = cosx/((x-2)(x+10))
The solution
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cos(x)
f(x) = ----------------
(x - 2)*(x + 10)
$$f{\left(x \right)} = \frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}$$
f = cos(x)/(((x - 2)*(x + 10)))
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -10$$
$$x_{2} = 2$$
$$x_{1} = -10$$
$$x_{2} = 2$$
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:
$$\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{2}$$
$$x_{2} = \frac{3 \pi}{2}$$
Numerical solution
$$x_{1} = 4.71238898038469$$
$$x_{2} = 17.2787595947439$$
$$x_{3} = -89.5353906273091$$
$$x_{4} = 64.4026493985908$$
$$x_{5} = 70.6858347057703$$
$$x_{6} = 36.1283155162826$$
$$x_{7} = -98.9601685880785$$
$$x_{8} = 48.6946861306418$$
$$x_{9} = -58.1194640914112$$
$$x_{10} = 7.85398163397448$$
$$x_{11} = 39.2699081698724$$
$$x_{12} = -95.8185759344887$$
$$x_{13} = -1.5707963267949$$
$$x_{14} = -92.6769832808989$$
$$x_{15} = 158.650429006285$$
$$x_{16} = -23.5619449019235$$
$$x_{17} = 23.5619449019235$$
$$x_{18} = 61.261056745001$$
$$x_{19} = 29.845130209103$$
$$x_{20} = -32.9867228626928$$
$$x_{21} = -51.8362787842316$$
$$x_{22} = -80.1106126665397$$
$$x_{23} = -83.2522053201295$$
$$x_{24} = 67.5442420521806$$
$$x_{25} = 98.9601685880785$$
$$x_{26} = 92.6769832808989$$
$$x_{27} = -39.2699081698724$$
$$x_{28} = -717.853921345268$$
$$x_{29} = 86.3937979737193$$
$$x_{30} = 45.553093477052$$
$$x_{31} = -67.5442420521806$$
$$x_{32} = 51.8362787842316$$
$$x_{33} = 76.9690200129499$$
$$x_{34} = -26.7035375555132$$
$$x_{35} = -4.71238898038469$$
$$x_{36} = 95.8185759344887$$
$$x_{37} = -86.3937979737193$$
$$x_{38} = -36.1283155162826$$
$$x_{39} = 83.2522053201295$$
$$x_{40} = -7.85398163397448$$
$$x_{41} = -14.1371669411541$$
$$x_{42} = -17.2787595947439$$
$$x_{43} = 20.4203522483337$$
$$x_{44} = 54.9778714378214$$
$$x_{45} = -70.6858347057703$$
$$x_{46} = -48.6946861306418$$
$$x_{47} = -54.9778714378214$$
$$x_{48} = -45.553093477052$$
$$x_{49} = -136.659280431156$$
$$x_{50} = 14.1371669411541$$
$$x_{51} = -73.8274273593601$$
$$x_{52} = 26.7035375555132$$
$$x_{53} = 89.5353906273091$$
$$x_{54} = 10.9955742875643$$
$$x_{55} = 174.358392274234$$
$$x_{56} = 359.712358836031$$
$$x_{57} = 80.1106126665397$$
$$x_{58} = 73.8274273593601$$
$$x_{59} = 58.1194640914112$$
$$x_{60} = -61.261056745001$$
$$x_{61} = -7858.69402295487$$
$$x_{62} = -20.4203522483337$$
$$x_{63} = -42.4115008234622$$
$$x_{64} = 32.9867228626928$$
$$x_{65} = 42.4115008234622$$
$$x_{66} = -76.9690200129499$$
$$x_{67} = -64.4026493985908$$
$$x_{68} = -29.845130209103$$
so we need to solve the equation:
$$\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{\pi}{2}$$
$$x_{2} = \frac{3 \pi}{2}$$
Numerical solution
$$x_{1} = 4.71238898038469$$
$$x_{2} = 17.2787595947439$$
$$x_{3} = -89.5353906273091$$
$$x_{4} = 64.4026493985908$$
$$x_{5} = 70.6858347057703$$
$$x_{6} = 36.1283155162826$$
$$x_{7} = -98.9601685880785$$
$$x_{8} = 48.6946861306418$$
$$x_{9} = -58.1194640914112$$
$$x_{10} = 7.85398163397448$$
$$x_{11} = 39.2699081698724$$
$$x_{12} = -95.8185759344887$$
$$x_{13} = -1.5707963267949$$
$$x_{14} = -92.6769832808989$$
$$x_{15} = 158.650429006285$$
$$x_{16} = -23.5619449019235$$
$$x_{17} = 23.5619449019235$$
$$x_{18} = 61.261056745001$$
$$x_{19} = 29.845130209103$$
$$x_{20} = -32.9867228626928$$
$$x_{21} = -51.8362787842316$$
$$x_{22} = -80.1106126665397$$
$$x_{23} = -83.2522053201295$$
$$x_{24} = 67.5442420521806$$
$$x_{25} = 98.9601685880785$$
$$x_{26} = 92.6769832808989$$
$$x_{27} = -39.2699081698724$$
$$x_{28} = -717.853921345268$$
$$x_{29} = 86.3937979737193$$
$$x_{30} = 45.553093477052$$
$$x_{31} = -67.5442420521806$$
$$x_{32} = 51.8362787842316$$
$$x_{33} = 76.9690200129499$$
$$x_{34} = -26.7035375555132$$
$$x_{35} = -4.71238898038469$$
$$x_{36} = 95.8185759344887$$
$$x_{37} = -86.3937979737193$$
$$x_{38} = -36.1283155162826$$
$$x_{39} = 83.2522053201295$$
$$x_{40} = -7.85398163397448$$
$$x_{41} = -14.1371669411541$$
$$x_{42} = -17.2787595947439$$
$$x_{43} = 20.4203522483337$$
$$x_{44} = 54.9778714378214$$
$$x_{45} = -70.6858347057703$$
$$x_{46} = -48.6946861306418$$
$$x_{47} = -54.9778714378214$$
$$x_{48} = -45.553093477052$$
$$x_{49} = -136.659280431156$$
$$x_{50} = 14.1371669411541$$
$$x_{51} = -73.8274273593601$$
$$x_{52} = 26.7035375555132$$
$$x_{53} = 89.5353906273091$$
$$x_{54} = 10.9955742875643$$
$$x_{55} = 174.358392274234$$
$$x_{56} = 359.712358836031$$
$$x_{57} = 80.1106126665397$$
$$x_{58} = 73.8274273593601$$
$$x_{59} = 58.1194640914112$$
$$x_{60} = -61.261056745001$$
$$x_{61} = -7858.69402295487$$
$$x_{62} = -20.4203522483337$$
$$x_{63} = -42.4115008234622$$
$$x_{64} = 32.9867228626928$$
$$x_{65} = 42.4115008234622$$
$$x_{66} = -76.9690200129499$$
$$x_{67} = -64.4026493985908$$
$$x_{68} = -29.845130209103$$
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
$$\frac{\left(- 2 x - 8\right) \cos{\left(x \right)}}{\left(x - 2\right)^{2} \left(x + 10\right)^{2}} - \frac{1}{\left(x - 2\right) \left(x + 10\right)} \sin{\left(x \right)} = 0$$
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
$$\frac{\left(- 2 x - 8\right) \cos{\left(x \right)}}{\left(x - 2\right)^{2} \left(x + 10\right)^{2}} - \frac{1}{\left(x - 2\right) \left(x + 10\right)} \sin{\left(x \right)} = 0$$
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
$$\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 45.4710070173758$$
$$x_{2} = -86.3449576389006$$
$$x_{3} = -42.3043930998253$$
$$x_{4} = -26.5119224835866$$
$$x_{5} = -83.2014041049884$$
$$x_{6} = -58.0445093013385$$
$$x_{7} = -190.044832546368$$
$$x_{8} = -89.4883646207564$$
$$x_{9} = 29.7225546767569$$
$$x_{10} = -67.4806517839074$$
$$x_{11} = -39.1526319974029$$
$$x_{12} = 54.9092447837187$$
$$x_{13} = -76.9137795411141$$
$$x_{14} = -70.6252971266199$$
$$x_{15} = 42.3236490425477$$
$$x_{16} = 89.4924261335424$$
$$x_{17} = -92.6316409673177$$
$$x_{18} = 64.3436583352986$$
$$x_{19} = 23.408490381794$$
$$x_{20} = 7.35769984295323$$
$$x_{21} = 76.9193095879542$$
$$x_{22} = -29.6801615396439$$
$$x_{23} = -32.8416196600561$$
$$x_{24} = 92.6354273858147$$
$$x_{25} = -13.3916037185683$$
$$x_{26} = 95.7783388455168$$
$$x_{27} = 51.7636909463359$$
$$x_{28} = -51.7511406381892$$
$$x_{29} = -64.3356786510842$$
$$x_{30} = 26.5673194144507$$
$$x_{31} = -16.8781399543188$$
$$x_{32} = 80.062780911349$$
$$x_{33} = 17.071740289959$$
$$x_{34} = -20.1312433663008$$
$$x_{35} = 83.2061146839006$$
$$x_{36} = 98.9211688644511$$
$$x_{37} = -4.80963959200728$$
$$x_{38} = 32.8752374467871$$
$$x_{39} = -61.1903212799701$$
$$x_{40} = 20.2443394322777$$
$$x_{41} = 86.3493255488836$$
$$x_{42} = -23.3323622844516$$
$$x_{43} = -98.9178546297908$$
$$x_{44} = -54.8981537926459$$
$$x_{45} = 39.1754015123576$$
$$x_{46} = 70.6318829945278$$
$$x_{47} = 61.1991721850654$$
$$x_{48} = -35.9986561408559$$
$$x_{49} = -80.0576853892703$$
$$x_{50} = 58.0543844350502$$
$$x_{51} = 73.7756836327888$$
$$x_{52} = -95.7748003609166$$
$$x_{53} = 1412.14307316361$$
$$x_{54} = 48.6176448144202$$
$$x_{55} = -48.6033207731632$$
$$x_{56} = -73.7696605087807$$
$$x_{57} = 10.6659557066639$$
$$x_{58} = -1.10871099362239$$
$$x_{59} = 36.0260362022415$$
$$x_{60} = -45.4544954337464$$
$$x_{61} = 67.4878839716098$$
$$x_{62} = 13.884286370083$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -10$$
$$x_{2} = 2$$
$$\lim_{x \to -10^-}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = -\infty$$
$$\lim_{x \to -10^+}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = -10$$
- is an inflection point
$$\lim_{x \to 2^-}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = \infty$$
$$\lim_{x \to 2^+}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = -\infty$$
- the limits are not equal, so
$$x_{2} = 2$$
- is an inflection 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
$$\left[95.7783388455168, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -98.9178546297908\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 45.4710070173758$$
$$x_{2} = -86.3449576389006$$
$$x_{3} = -42.3043930998253$$
$$x_{4} = -26.5119224835866$$
$$x_{5} = -83.2014041049884$$
$$x_{6} = -58.0445093013385$$
$$x_{7} = -190.044832546368$$
$$x_{8} = -89.4883646207564$$
$$x_{9} = 29.7225546767569$$
$$x_{10} = -67.4806517839074$$
$$x_{11} = -39.1526319974029$$
$$x_{12} = 54.9092447837187$$
$$x_{13} = -76.9137795411141$$
$$x_{14} = -70.6252971266199$$
$$x_{15} = 42.3236490425477$$
$$x_{16} = 89.4924261335424$$
$$x_{17} = -92.6316409673177$$
$$x_{18} = 64.3436583352986$$
$$x_{19} = 23.408490381794$$
$$x_{20} = 7.35769984295323$$
$$x_{21} = 76.9193095879542$$
$$x_{22} = -29.6801615396439$$
$$x_{23} = -32.8416196600561$$
$$x_{24} = 92.6354273858147$$
$$x_{25} = -13.3916037185683$$
$$x_{26} = 95.7783388455168$$
$$x_{27} = 51.7636909463359$$
$$x_{28} = -51.7511406381892$$
$$x_{29} = -64.3356786510842$$
$$x_{30} = 26.5673194144507$$
$$x_{31} = -16.8781399543188$$
$$x_{32} = 80.062780911349$$
$$x_{33} = 17.071740289959$$
$$x_{34} = -20.1312433663008$$
$$x_{35} = 83.2061146839006$$
$$x_{36} = 98.9211688644511$$
$$x_{37} = -4.80963959200728$$
$$x_{38} = 32.8752374467871$$
$$x_{39} = -61.1903212799701$$
$$x_{40} = 20.2443394322777$$
$$x_{41} = 86.3493255488836$$
$$x_{42} = -23.3323622844516$$
$$x_{43} = -98.9178546297908$$
$$x_{44} = -54.8981537926459$$
$$x_{45} = 39.1754015123576$$
$$x_{46} = 70.6318829945278$$
$$x_{47} = 61.1991721850654$$
$$x_{48} = -35.9986561408559$$
$$x_{49} = -80.0576853892703$$
$$x_{50} = 58.0543844350502$$
$$x_{51} = 73.7756836327888$$
$$x_{52} = -95.7748003609166$$
$$x_{53} = 1412.14307316361$$
$$x_{54} = 48.6176448144202$$
$$x_{55} = -48.6033207731632$$
$$x_{56} = -73.7696605087807$$
$$x_{57} = 10.6659557066639$$
$$x_{58} = -1.10871099362239$$
$$x_{59} = 36.0260362022415$$
$$x_{60} = -45.4544954337464$$
$$x_{61} = 67.4878839716098$$
$$x_{62} = 13.884286370083$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -10$$
$$x_{2} = 2$$
$$\lim_{x \to -10^-}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = -\infty$$
$$\lim_{x \to -10^+}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = -10$$
- is an inflection point
$$\lim_{x \to 2^-}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = \infty$$
$$\lim_{x \to 2^+}\left(\frac{- \cos{\left(x \right)} + \frac{4 \left(x + 4\right) \sin{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} + \frac{2 \left(\left(x + 4\right) \left(\frac{1}{x + 10} + \frac{1}{x - 2}\right) + \frac{x + 4}{x + 10} - 1 + \frac{x + 4}{x - 2}\right) \cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = -\infty$$
- the limits are not equal, so
$$x_{2} = 2$$
- is an inflection 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
$$\left[95.7783388455168, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -98.9178546297908\right]$$
Vertical asymptotes
Have:
$$x_{1} = -10$$
$$x_{2} = 2$$
$$x_{1} = -10$$
$$x_{2} = 2$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)/(((x - 2)*(x + 10))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{\left(x - 2\right) \left(x + 10\right)} \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{1}{\left(x - 2\right) \left(x + 10\right)} \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{\left(x - 2\right) \left(x + 10\right)} \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{1}{\left(x - 2\right) \left(x + 10\right)} \cos{\left(x \right)}}{x}\right) = 0$$
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:
$$\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} = \frac{\cos{\left(x \right)}}{\left(10 - x\right) \left(- x - 2\right)}$$
- No
$$\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} = - \frac{\cos{\left(x \right)}}{\left(10 - x\right) \left(- x - 2\right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} = \frac{\cos{\left(x \right)}}{\left(10 - x\right) \left(- x - 2\right)}$$
- No
$$\frac{\cos{\left(x \right)}}{\left(x - 2\right) \left(x + 10\right)} = - \frac{\cos{\left(x \right)}}{\left(10 - x\right) \left(- x - 2\right)}$$
- No
so, the function
not is
neither even, nor odd