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 equationThe 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
$$\left(-\infty, -98.9178546297908\right]$$