Weird natural log question

**You are given the function**

f(x) = x*lnx

(x > 0)

a) Find the solution(s) of the equation f(x) = 0

b) Find any minima/maxima that f(x) has.

c) Find the value(s) of x0 for which

INTEGRAL ( between limits x0 and 0) f(x) dx = 0

and describe graphically what this means.

Ok first of all, for part a) all I can think of for a solution is x = 1 because lnx = 0 only when x = 1, right? Thing is this question is worth 4 marks so I feel like im missing something.

Secondly for part b) I didnt think there were any maxima/minima because of what I said above. However, differentiating the function I get

lnx + x(1/x)

equating to zero gives

lnx + 1 = 0

lnx = -1

lnx = -lne

lnx = lne^-1

x = 1/e

Plugging this into the second derivative (which is 1/x) gets me e as a minima.

Is this correct?

And for the last part it tells me to use integration by parts, but it doesnt get me anywhere...