Consider the function

$$f(x) = \sin x + \frac{1}{\sqrt{2}} \cos 2x$$

where $x \in (0, 2\pi)$. Find the extreme values of the function in the interval $x \in (0, 2\pi)$:
a. Find function $f'(x)$
b. Find $f''(x)$
c. Find the values of $x \in (0, 2\pi)$ for which $f(x)$ has maximum or minimum values
d. Find possible extreme values of $f(x)$.