Solve the equation x-sinx=0.
In order to solve the equation, which is a transcedental one, you have to find the derivative of the function f.
Yes, we can find the derivative of the function f, because is a continuous function, formed by elementary functions as the linear one,x , and the trigonometrical one, sin x!
We can note that is a monotone increasing function
( -1<cosx<1), so the difference 1 -cos x>0 =>f(x)>0, so f(x) is an injection.
We can also do a very simple calculus:
We've demonstrated earlier before that f(x) is an injection, so, x=0 is the only solution for x-sinx=0