I have to determine global and local extrema of the above function which is defined in the interval $[0,e]$ and to determine the global ones I need to check the value in $0$ and in $e$ right?
In $e$ the value is 1, that's obvious, but what do I do with zero? I looked at a graph and apparently $f(0)=1$, but obviously $\ln(0)$ is not defined since $f(x) \to -\infty$ as $x \to 0$.
I'm a bit clueless as to what to do now so I would really appreciate it if someone explained what I should do
You seem to forget that $$\lim_{x\to 0^+} \, x \log(x)=0$$ For the remaining, consider the function $$f(x)=x \log (x)-x+1$$ Compute its first derivative, check where it does cancels and use the second derivative test to see if this corresponds to a maximum or a minimum.