I want to know if my solution for this problem is correct or if I am doing anything wrong.
I have to prove that the equation $2+\frac{x}{2}+\ln(x)=0$ has a solution in the intervall $I=\left(0, \tfrac{1}{e}\right)$.
so I have to test if the function $f(x) =2+\frac{x}{2}+\ln(x) $ has a root in $I$.
With $\displaystyle\lim_{x \rightarrow 0}f(x)=- \infty$ and $\displaystyle\lim_{x \rightarrow \tfrac{1}{e}}f(x)=\left(1+\tfrac{1}{2e}\right)>0$ We see that the function has at least one root.
With the first derivative $f'(x)=\frac{1}{2}+\frac{1}{x}$ we see that, since the first derivative is positive and monotonically decreasing in $I$, the function has only one root. (The function can only go up since there is no point where the slope becomes negative).
Am I doing anything wrong?