Question

Using the principle of the mean-value theorem on the indicated interval, how do you find all numbers c that satisfy the conclusion of the theorem `f(x)=x-cosx`; [-pi/2, pi/2]?

Answer 1

See below.

Explanation:

I am unaware of anything that might be called "the principle of the MVT" that can be used to do this.

What we need is knowledge of what the conclusion of MVT says and enough algebra and trigonometry skills to solve an equation.

Solution

The conclusion of MVT for this function on this interval says:

there is a `c` in `(-pi/2,pi/2)` such that `f'(c) = (f(pi/2)-f(-pi/2))/(pi/2-(-pi/2))`

Now do the calculus, trigonometry and algebra to find `c`.

This amounts to solving

`1+sinx = 1` on the interval `(-pi/2,pi/2)`.

So, `c=0`