Inequality regarding exponential function

Question

For every positive $x$ and for every $n$ show that $(1+\sqrt{\frac{x}{n}}+\sqrt[3]{\frac{x}{n}})e^{n\arctan{\frac{x}{n}}}>e^x$.

I plotted it, it seems that the inequality holds. Any ideas how to prove this?

Answer 1

The inequality $$(1+\sqrt{\frac{x}{n}}+\sqrt[3]{\frac{x}{n}})e^{n\arctan{\frac{x}{n}}}>e^x$$ only holds for a limited range $0<x<r_n$ where $r_n$ is the solution of $$f(x,n)=(1+\sqrt{\frac{x}{n}}+\sqrt[3]{\frac{x}{n}})e^{n\arctan{\frac{x}{n}}}-e^x=0$$ Numerically $$r_1=2.58234$$ $$r_2=3.29207$$ $$r_3=3.93284$$ $$r_4=4.51660$$ $$r_5=5.05721$$