Use the Intermediate Value Theorem and Mean Value Theorem to show that the queation $2x-1-sinx=0$ has exactly one root.
My answer :
Since we cannot compute the $y$ when $x=0$, we use the Intermediate Value Theorem.
Randomly choose two points : When x = -10, 2(-10) -1 -sin(-10) = -21.5, (-10, -21.5) When x = 100, 2(100) -1 -sin(100) = 199.51, (100, 199.51)
Since $y=2x-1 -sin(x)$ is a continuous curve and the two points exist at different sign regions, there is at least one point at $y=0$.
For the Mean Value Theorem, how to use it for the proof?
Thank you for your help.