Mathematics

Problem

Consider the function

g(x)=\sin x+\cos 2x

on the interval

[0,\pi]

. (i) Calculate the average value of

g(x)

over

[0,\pi]

. (ii) Prove that there exists at least one number

c

(0,\pi)

satisfying

g(c)

equal to the average value found in part (i), and determine one such value of

c

correct to two decimal places.