Given

$$f(x)=\left\{

\begin{array}{ll}

x\cdot \ln(x) & \mbox{if } x > 0 \\

0 & \mbox{if } x = 0

\end{array}

\right.

$$

a)Prove that $f(x)$ is continuous at $x=0$.

b)Study $f(x)$ as for the monotony and find its domain.

Find the number of the positive roots of the equation $x=e^{a/x}$ for all the real values of $a$.

c)

d)Prove: $f'(x+1)>f(x+1)-f(x)$, $\forall x>0$.

Questions

Note:a, bhave already been solved. I'm struggling to move on to questionscandd.

it is continuous at 0.a)

f(Df)=$[-(1/e),\infty)$, at $(0,1/e]$ f(x) is a strictly decreasing function whereas at $[1/e,\infty)$ f(x) is a strictly increasing function.b)

RegardingcI've thought of usingbut not exactly sure on how to apply it in the current situation.Bolzano's Rule