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?
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 $x$.
