Need help with a mean value theorem problem

Question

Does the Mean value theorem apply to the given function on the given interval? Is so, find "c". If not, why not?

f(x) = x - sin(pi x), [-1,1]

Answer 1

f(x) = x - sin(πx) is continuous on [-1,1] and is differentiable on (-1,1).  So, the Mean Value Theorem does apply.  

 

By the Mean Value Theorem, there exists at least one number, c, in the interval (-1,1) for which f'(c) = [f(1) - f(-1)]/(1- (-1))

 

So, 1 - πcos(πc) = (1- (-1))/2

 

     1 - πcos(πc) = 1

 

            cos(πc) = 0     πc = π/2 + kπ    k = 0, ±1, ±2, ...

 

                                  c = 1/2 + k

 

The only solutions in the interval (-1,1) are c = 1/2 and -1/2.