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$$7^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(7 \right)}^{2} + 2 \cdot 7^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(7 \right)} \tan{\left(x \right)} - 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -62.3769176845804$$
$$x_{2} = -98.9601685880785$$
$$x_{3} = 53.8620104982419$$
$$x_{4} = 92.6769832808989$$
$$x_{5} = -48.6946861306418$$
$$x_{6} = -100.076029527658$$
$$x_{7} = -64.4026493985908$$
$$x_{8} = -12.1114352271437$$
$$x_{9} = 91.5611223413195$$
$$x_{10} = 73.8274273593601$$
$$x_{11} = 45.553093477052$$
$$x_{12} = 89.5353906273091$$
$$x_{13} = 31.8708619231134$$
$$x_{14} = 76.9690200129499$$
$$x_{15} = -93.7928442204783$$
$$x_{16} = 60.1451958054215$$
$$x_{17} = 63.2867884590113$$
$$x_{18} = -58.1194640914112$$
$$x_{19} = 88.4195296877297$$
$$x_{20} = -61.261056745001$$
$$x_{21} = -51.8362787842316$$
$$x_{22} = 3.59652804080525$$
$$x_{23} = 7.85398163397448$$
$$x_{24} = 58.1194640914112$$
$$x_{25} = 23.5619449019235$$
$$x_{26} = -46.6689544166314$$
$$x_{27} = 75.8531590733705$$
$$x_{28} = -78.0848809525294$$
$$x_{29} = -34.1025838022723$$
$$x_{30} = -32.9867228626928$$
$$x_{31} = 82.1363443805501$$
$$x_{32} = -70.6858347057703$$
$$x_{33} = 25.5876766159338$$
$$x_{34} = -49.8105470702212$$
$$x_{35} = -95.8185759344887$$
$$x_{36} = -26.7035375555132$$
$$x_{37} = -68.66010299176$$
$$x_{38} = 85.2779370341399$$
$$x_{39} = 29.845130209103$$
$$x_{40} = -24.6778058415029$$
$$x_{41} = 80.1106126665397$$
$$x_{42} = 16.1628986551644$$
$$x_{43} = -76.9690200129499$$
$$x_{44} = 61.261056745001$$
$$x_{45} = 66.4283811126011$$
$$x_{46} = -80.1106126665397$$
$$x_{47} = 51.8362787842316$$
$$x_{48} = -29.845130209103$$
$$x_{49} = -5.82824991996413$$
$$x_{50} = -20.4203522483337$$
$$x_{51} = -65.5185103381702$$
$$x_{52} = -17.2787595947439$$
$$x_{53} = -43.5273617630417$$
$$x_{54} = 67.5442420521806$$
$$x_{55} = -90.6512515668885$$
$$x_{56} = 4.71238898038469$$
$$x_{57} = 70.6858347057703$$
$$x_{58} = -71.8016956453498$$
$$x_{59} = 32.9867228626928$$
$$x_{60} = -83.2522053201295$$
$$x_{61} = -36.1283155162826$$
$$x_{62} = -21.5362131879131$$
$$x_{63} = 9.87971334798483$$
$$x_{64} = 1.5707963267949$$
$$x_{65} = -92.6769832808989$$
$$x_{66} = -27.8193984950927$$
$$x_{67} = -87.5096589132988$$
$$x_{68} = 22.446083962344$$
$$x_{69} = 17.2787595947439$$
$$x_{70} = 10.9955742875643$$
$$x_{71} = 19.3044913087542$$
$$x_{72} = 95.8185759344887$$
$$x_{73} = -14.1371669411541$$
$$x_{74} = -40.3857691094519$$
$$x_{75} = 54.9778714378214$$
$$x_{76} = -4.71238898038469$$
$$x_{77} = 83.2522053201295$$
$$x_{78} = 14.1371669411541$$
$$x_{79} = 98.9601685880785$$
$$x_{80} = -84.368066259709$$
$$x_{81} = 69.5699737661909$$
$$x_{82} = 26.7035375555132$$
$$x_{83} = -18.3946205343233$$
$$x_{84} = 48.6946861306418$$
$$x_{85} = 41.2956398838828$$
$$x_{86} = -86.3937979737193$$
$$x_{87} = 97.844307648499$$
$$x_{88} = 44.4372325374726$$
$$x_{89} = 38.154047230293$$
$$x_{90} = 39.2699081698724$$
$$x_{91} = -7.85398163397448$$
$$x_{92} = 47.5788251910624$$
$$x_{93} = -39.2699081698724$$
$$x_{94} = -56.0937323774008$$
$$x_{95} = -10.9955742875643$$
$$x_{96} = -73.8274273593601$$
$$x_{97} = -2.68665726637434$$
$$x_{98} = -54.9778714378214$$
$$x_{99} = -42.4115008234622$$
$$x_{100} = 36.1283155162826$$
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
$$\left[98.9601685880785, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.076029527658\right]$$