Question

Given the function `f(x)=x-cosx`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-pi/2,pi/2] and find the c?

Answer 1

You determine whether it satisfies the hypotheses by determining whether `f(x) = x-cosx` is continuous on the interval `[-pi/2,pi/2]` and differentiable on the interval `(-pi/2,pi/2)`. (Those are the hypotheses of the Mean Value Theorem)

You find the `c` mentioned in the conclusion of the theorem by solving `f'(x) = (f(pi/2)-f(-pi/2))/(pi/2-(-pi/2))` on the interval `(-pi/2,pi/2)`.

`f` is continuous on its domain, which includes `[-pi/2,pi/2]`

`f'(x) = 1+sinx` which exists for all `x` so it exists for all `x` in `(-pi/2,pi/2)`

Therefore this function satisfies the hypotheses of the Mean Value Theorem on this interval.

To find `c` solve the equation `f'(x) = (f(pi/2)-f(-pi/2))/(pi/2-(-pi/2))` . Discard any solutions outside `(-pi/2,pi/2)`.

You should get `c = 0`.