4)a) prove that $\displaystyle g(x)=e^{x}+\frac{x^{3}}{x^{2}+1} $gets a unique value for every x in R

4)b)prove that $\displaystyle f(x)=e^{x}+\frac{x^{2}}{2}-\frac{ln(x^{2}+1)}{2}$ gets total minimum (not only local)

you can use part 4a)

4)c)prove that $\displaystyle \frac{ln(x+1)}{x}$ 'uniformly' continuous in $\displaystyle (0,\infty)$