El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
$$\frac{2 x}{x x + 1} - \frac{2 x \left(x x - 1\right)}{\left(x x + 1\right)^{2}} = 0$$
Resolvermos esta ecuaciónPuntos de cruce con el eje X:
Solución analítica$$x_{1} = 0$$
Solución numérica$$x_{1} = 40754.0146073646$$
$$x_{2} = 22109.5852761254$$
$$x_{3} = 31431.3751377026$$
$$x_{4} = 16178.6518916848$$
$$x_{5} = 29736.4176408609$$
$$x_{6} = -16047.4885294021$$
$$x_{7} = -20283.7072775985$$
$$x_{8} = 21262.2305994173$$
$$x_{9} = 34821.3634468047$$
$$x_{10} = 42449.0875005062$$
$$x_{11} = 15331.5208433594$$
$$x_{12} = -15200.3654382736$$
$$x_{13} = -36385.1680059945$$
$$x_{14} = 39058.9534836425$$
$$x_{15} = 20414.8966506568$$
$$x_{16} = -23673.1469943384$$
$$x_{17} = -24520.549154346$$
$$x_{18} = -41470.3270985658$$
$$x_{19} = -39775.2608173139$$
$$x_{20} = 32278.8637725529$$
$$x_{21} = -32995.141812677$$
$$x_{22} = 18720.3022019605$$
$$x_{23} = -38080.207196563$$
$$x_{24} = 36516.3874515588$$
$$x_{25} = -38927.7323209959$$
$$x_{26} = -21131.0378020927$$
$$x_{27} = 24651.7522903592$$
$$x_{28} = 39906.4824805783$$
$$x_{29} = 30583.8929111256$$
$$x_{30} = 33973.8583483916$$
$$x_{31} = -16894.6599134202$$
$$x_{32} = -37232.6856743107$$
$$x_{33} = 19567.5861240751$$
$$x_{34} = 17873.0486704095$$
$$x_{35} = 35668.8732572414$$
$$x_{36} = 27194.0400733442$$
$$x_{37} = -21978.3894419472$$
$$x_{38} = -31300.1604633294$$
$$x_{39} = -35537.6544673896$$
$$x_{40} = 28041.4904927672$$
$$x_{41} = -22825.7598445709$$
$$x_{42} = -17741.8726673034$$
$$x_{43} = 25499.1699170213$$
$$x_{44} = 33126.3583237319$$
$$x_{45} = -25367.9648198459$$
$$x_{46} = -30452.6792719769$$
$$x_{47} = -28757.7386512816$$
$$x_{48} = -42317.8645113938$$
$$x_{49} = 38211.4278248492$$
$$x_{50} = -40622.792474443$$
$$x_{51} = 23804.3479553544$$
$$x_{52} = -32147.6481435989$$
$$x_{53} = 28888.9499392717$$
$$x_{54} = 26346.5995523547$$
$$x_{55} = 41601.5496726855$$
$$x_{56} = 22956.9583845$$
$$x_{57} = -29605.2051269265$$
$$x_{58} = -34690.1453612251$$
$$x_{59} = -27062.8315909668$$
$$x_{60} = -19436.4006315048$$
$$x_{61} = -27910.2805438436$$
$$x_{62} = -33842.6410205595$$
$$x_{63} = 17025.8300696591$$
$$x_{64} = 0$$
$$x_{65} = 37363.9057313917$$
$$x_{66} = -26215.3926807132$$
$$x_{67} = -18589.1211308922$$