The question is asking about the conditions under which the Fourier series of a function $f(x)$ in the interval $(0, 2\pi)$ is possible. The options are:

(a) $f(x)$ is an even function
(b) $f(x)$ is an odd function
(c) $f(x)$ is a periodic function
(d) All of these

Explanation:

The Fourier series is used to represent periodic functions as a sum of sines and cosines. For a Fourier series to exist, the function must satisfy certain conditions:

• Periodic Function (Option c): The function $f(x)$ must be periodic. This is a fundamental requirement since the Fourier series is specifically designed for periodic functions. The period of the function is generally considered to be $2\pi$ for such cases.

• Even or Odd Functions (Options a and b): Whether the function is even or odd affects the specific form of the Fourier series (cosine terms for even functions and sine terms for odd functions). However, the function doesn't necessarily need to be even or odd for the Fourier series to exist. These are special cases where the series simplifies, but the general condition is periodicity.

Therefore, the Fourier series is possible when $f(x)$ is a periodic function (Option c). The function does not need to be strictly even or odd, so the correct answer is (c) $f(x)$ is a periodic function.

Would you like more details on Fourier series conditions or any other clarification?

Here are five related questions you might explore:

1. What is the Fourier series of a non-periodic function?
2. How does the Fourier series change for an even vs. odd function?
3. Can a discontinuous function have a Fourier series representation?
4. What is the difference between the Fourier transform and the Fourier series?
5. How are the coefficients of a Fourier series calculated?

Tip: A function need not be even or odd to have a Fourier series. Periodicity is the key requirement.