Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada$$- 18 \left(\left(\tan^{2}{\left(3 x \right)} + 1\right) \sin{\left(\tan{\left(3 x \right)} \right)} - 2 \cos{\left(\tan{\left(3 x \right)} \right)} \tan{\left(3 x \right)}\right) \left(\tan^{2}{\left(3 x \right)} + 1\right) = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 32.2447712509389$$
$$x_{2} = 36.4335614557253$$
$$x_{3} = -24.0855436775217$$
$$x_{4} = 40.0118597816263$$
$$x_{5} = -46.0766922526503$$
$$x_{6} = 80.4158586059824$$
$$x_{7} = -77.7109716247038$$
$$x_{8} = -65.7550928892301$$
$$x_{9} = 90.0589894029074$$
$$x_{10} = 78.3214635035892$$
$$x_{11} = -79.5870138909414$$
$$x_{12} = 8.37758040957278$$
$$x_{13} = -21.9911485751286$$
$$x_{14} = 4.18879020478639$$
$$x_{15} = 81.8997618294902$$
$$x_{16} = -91.106186954104$$
$$x_{17} = -81.6814089933346$$
$$x_{18} = -55.7198230495753$$
$$x_{19} = 46.0766922526503$$
$$x_{20} = -59.6902604182061$$
$$x_{21} = -94.0294267715382$$
$$x_{22} = 98.2182169763246$$
$$x_{23} = -41.8879020478639$$
$$x_{24} = 15.9263161041046$$
$$x_{25} = 43.9822971502571$$
$$x_{26} = -74.1326732988028$$
$$x_{27} = -39.7935069454707$$
$$x_{28} = -15.707963267949$$
$$x_{29} = 342.651952077443$$
$$x_{30} = -50.0471296212811$$
$$x_{31} = -9.64313079692497$$
$$x_{32} = -13.3952153294002$$
$$x_{33} = -96.1238218739314$$
$$x_{34} = 0$$
$$x_{35} = -75.6165765223106$$
$$x_{36} = -32.6814769232501$$
$$x_{37} = 19.8967534727354$$
$$x_{38} = -28.0559810461525$$
$$x_{39} = 34.3391663533321$$
$$x_{40} = 12.3480177782036$$
$$x_{41} = -6.06483247102399$$
$$x_{42} = 41.8879020478639$$
$$x_{43} = -47.9527345188879$$
$$x_{44} = 63.8790506229925$$
$$x_{45} = 68.0678408277789$$
$$x_{46} = 74.3510261349584$$
$$x_{47} = -99.7021201998324$$
$$x_{48} = -90.0589894029074$$
$$x_{49} = -31.6342793720535$$
$$x_{50} = 28.2743338823081$$
$$x_{51} = -72.0382781964096$$
$$x_{52} = 65.9734457253857$$
$$x_{53} = -85.870199198121$$
$$x_{54} = -61.7846555205993$$
$$x_{55} = 83.9941569318834$$
$$x_{56} = -3.9704373686308$$
$$x_{57} = -11.7375258993182$$
$$x_{58} = -2.0943951023932$$
$$x_{59} = 58.4247100308539$$
$$x_{60} = 26.1799387799149$$
$$x_{61} = 52.3598775598299$$
$$x_{62} = -69.9438830940165$$
$$x_{63} = -53.6254279471821$$
$$x_{64} = -29.7551998380728$$
$$x_{65} = 59.9086132543617$$
$$x_{66} = -87.7462414643586$$
$$x_{67} = 96.342174710087$$
$$x_{68} = 13.3952153294002$$
$$x_{69} = -19.8967534727354$$
$$x_{70} = -68.0678408277789$$
$$x_{71} = 37.9174646792331$$
$$x_{72} = 56.3303149284607$$
$$x_{73} = 85.870199198121$$
$$x_{74} = 94.2477796076938$$
$$x_{75} = -97.171019425128$$
$$x_{76} = -52.3598775598299$$
$$x_{77} = -25.9615859437594$$
$$x_{78} = 72.2566310325652$$
$$x_{79} = 62.0030083567549$$
$$x_{80} = -83.7758040957278$$
$$x_{81} = -58.6430628670095$$
$$x_{82} = 70.162235930172$$
$$x_{83} = -17.8023583703422$$
$$x_{84} = -7.54873569453178$$
$$x_{85} = 2.0943951023932$$
$$x_{86} = 48.1710873550435$$
$$x_{87} = 24.0855436775217$$
$$x_{88} = 54.2359198260675$$
$$x_{89} = 6.28318530717959$$
$$x_{90} = -33.7286744744467$$
$$x_{91} = 21.9911485751286$$
$$x_{92} = 87.9645943005142$$
$$x_{93} = 76.227068401196$$
$$x_{94} = -63.8790506229925$$
$$x_{95} = 50.2654824574367$$
$$x_{96} = 92.1533845053006$$
$$x_{97} = -37.6991118430775$$
$$x_{98} = 18.0207112064978$$
$$x_{99} = 10.2536226758104$$
$$x_{100} = 30.3687289847013$$
$$x_{101} = 100.312612078718$$
Intervalos de convexidad y concavidad:Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[96.342174710087, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -91.106186954104\right]$$