For what value of ##k## will ##x+k/x## have a relative maximum at ##x=2##?

There is a relative minimum when ##x=2## with ##k=4##, but there is no possible value of k which gives a relative maximum when ##x=2##

Let

##f(x) = x+k/x##

Then differentiating wrt ##x## we get

## f'(x) = 1 - k/x^2 ##

And differentiating again wrt ##x## we get:

## f''(x) = (2k)/x^3 ##

the ##f'(2) = 1-k/4##

At a maximum or minimum we require ##f'(x) =0##, so for a maximum when ##x=2## we must have ##f'(2)=0##

## f'(2)=0 => 1 - k/2^2=0## ## :. 1-k/4 = 0## ## :. k = 4##

So When ##k=4 => f'(x)=0## when ##x=2## giving a single critical point

Now let's find the nature of this critical point. With ##k=4## and ##x=2##

## f''(2) = ((2)(4))/2^3 > 0 ##, Hence this a relative minimum