Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$e^{- x} \sqrt[3]{x} = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = 0$$
Numerical solution$$x_{1} = 33.5286981570306$$
$$x_{2} = 29.6269095273327$$
$$x_{3} = 39.4452332453898$$
$$x_{4} = 65.3133253157007$$
$$x_{5} = 111.252069207491$$
$$x_{6} = 93.2676977460751$$
$$x_{7} = 89.2721674640506$$
$$x_{8} = 81.2826726466621$$
$$x_{9} = 119.246814492279$$
$$x_{10} = 45.3951328784198$$
$$x_{11} = 35.4951987253783$$
$$x_{12} = 51.3613914384959$$
$$x_{13} = 75.2923193654619$$
$$x_{14} = 103.258249611654$$
$$x_{15} = 105.256604588073$$
$$x_{16} = 115.249339684993$$
$$x_{17} = 79.2856943090103$$
$$x_{18} = 85.277130127622$$
$$x_{19} = 121.245621461622$$
$$x_{20} = 0$$
$$x_{21} = 91.2698757192509$$
$$x_{22} = 61.3241781954468$$
$$x_{23} = 57.3370136179432$$
$$x_{24} = 107.255029001296$$
$$x_{25} = 97.2636506804946$$
$$x_{26} = 95.2656252377379$$
$$x_{27} = 113.250677380766$$
$$x_{28} = 71.2998543086375$$
$$x_{29} = 49.3713562973095$$
$$x_{30} = 53.3524413315494$$
$$x_{31} = 55.3443558773275$$
$$x_{32} = 101.259968785345$$
$$x_{33} = 99.2617672608143$$
$$x_{34} = 87.2745821883794$$
$$x_{35} = 83.279822693014$$
$$x_{36} = 67.3085026225693$$
$$x_{37} = 77.288903806042$$
$$x_{38} = 43.4094921706822$$
$$x_{39} = 109.253518530562$$
$$x_{40} = 31.5710744304598$$
$$x_{41} = 47.3825238369606$$
$$x_{42} = 37.4679295763206$$
$$x_{43} = 73.2959616736576$$
$$x_{44} = 41.4260089466185$$
$$x_{45} = 69.3040242686795$$
$$x_{46} = 59.3303151318917$$
$$x_{47} = 117.248053013168$$
$$x_{48} = 63.3185341779381$$