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...