# Thread: Find all the positive roots of a given equation and prove an inequality w/derivatives

1. ## Find all the positive roots of a given equation and prove an inequality w/derivatives

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.

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

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

Note:
Questions a, b have already been solved. I'm struggling to move on to questions c and d.
a) it is continuous at 0.
b) 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.
Regarding c I've thought of using Bolzano's Rule but not exactly sure on how to apply it in the current situation.

2. ## Re: Find all the positive roots of a given equation and prove an inequality w/derivat

(c) this is equivalent to solving

$$\ln x = \frac{a}{x}$$

or

$$f(x)=a$$

3. ## Re: Find all the positive roots of a given equation and prove an inequality w/derivat

which part? taking $ln$ of both sides we get $x ln x=a$ which is $f(x)=a$ for $x>0$
so we are asked to find the solutions of $f(x)=a$
we can use the graph of $f$ to answer this question