# Thread: Weird natural log question

1. ## 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...

2. Originally Posted by BogStandard
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.
if x*ln(x) = 0 then either x = 0 or ln(x) = 0 because if you have a product of two terms that yield zero, one or the other must be zero. however, we have that x > 0 (we need this for the log to be defined) so, you are correct, you are left with only one solution, x = 1.

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
good!

Plugging this into the second derivative (which is 1/x) gets me e as a minima.
Is this correct?
yup. you're a genius!

And for the last part it tells me to use integration by parts, but it doesnt get me anywhere...
use integration by parts with $u = \ln x$ and $dv = x$

you know what u and dv represent, right?