Q:

Sketch $f(x)=\sin x+\frac{1}{x}$ and local maxima and minima, intervals of increase and decrease,

Sketch $f(x)=\sin x+\frac{1}{x}$ finding local maxima and minima, intervals of increase and decrease.
I'm trying to use differentiation to draw this picture and find critical points.
So, I get $f'(x)=\cos x-\frac{1}{x^2}$
However, I'll have to deal with inequality $f'(x)>0 $, $f'(x)<0$, and  $f'(x)=0$, I feel I lack an ability to solve an equation like this. So is there any better way to find local maximun and minumum for this function?

A:

For $f'$,
note that,
if $|x| < 1$,
then
$f'(x) \ne 0$
(do you see why?).
Also, for large $x$,
since
$\cos(x)
=\sin(\pi/2-x)
$,
you want
$\sin(\pi/2-x)$
small.
Therefore,
$\pi/2-x$
has to be close
to a multiple of
$\pi$,
since that is when $\sin$ is small.
So,
let $\pi/2 -x=\pi n +z$,
where
$z$ is small.
Then
$\cos(x)
=\sin(\pi/2-x)
=\sin(\pi n+z)
\approx (-1)^n z
$,
so
$\frac1{x^2}
\approx (-1)^n z
\approx (-1)^n (\pi/2-\pi n)-x
$.
Since
$x$ is large,
$x
\approx (-1)^n (\pi/2-\pi n)
$.
Plot $f$ and $f'$
and see if these remarks
agree with what you see.
Remember,
the most useful approximation
for trig functions is
$\sin(x) \approx x$
for small $x$.

