Question

find all values of x satisfying the mean value theorem for f(x)=x-2 sinx

Answer 1

The MVT states that given a function that is continuous over \([a,b]\) and differentiable over \((a,b)\), then there is a value \(c\) between \(a\) and \(b\) such that the slope of the tangent to the function at \(c\) is the same as the slope of the line between the endpoints:
\[f'(c)=\frac{f(b)-f(a)}{b-a}\]
Where's your interval? We're missing information.