Question

What is the local max of f(x) =x sin x?

Answer 1

`x~~+-8 or x~~+-2`

For more...
https://www.wolframalpha.com/input/?i=max(xsin(x))

Explanation:

A maxima at point `a` must have `f'(a)=0` and `f''(a)<0`

`f'(x)=color(blue)(sin(x))+color(red)(xcos(x))`
`f''(x)=color(blue)(cos(x))+color(red)(cos(x)-xsin(x))=2cos(x)-xsin(x)`

If you want to have the derivatives explaint, just write a comment or send a note.

`f'(x)=0=sin(x)+xcos(x)`

The following lines can only be calculated by approximation e.g. a calculator.

`x~~+-8`
`f''(+-8)<0`
This is a max

`x~~+-5`
`f''(+-5)>0`

`x~~+-2`
`f''(+-2)<0`
This is a max

`x=0`
`f''(0)=1>0`

Checking by drawing the graph
graph{xsin(x) [-20, 20, -20, 20]}