Question

The following function is defined over the interval \([-L, L]\): \[ f(x) = px^4 + qx^5 \] If it is expressed as a Fourier series, \[ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \sin \left( \frac{n \pi x}{L} \right) + b_n \cos \left( \frac{n \pi x}{L} \right) \right), \] which options amongst the following are true?

• A. \( a_n, n = 1, 2, \ldots, \infty \) depend on \( p \)
• B. \( a_n, n = 1, 2, \ldots, \infty \) depend on \( q \)
• C. \( b_n, n = 1, 2, \ldots, \infty \) depend on \( p \)
• D. \( b_n, n = 1, 2, \ldots, \infty \) depend on \( q \)

Hint:

In Fourier series, \( a_n \) corresponds to the sine terms (odd powers of \( x \)) and \( b_n \) corresponds to the cosine terms (even powers of \( x \)).

Answer 1

The Correct Option is B, C

Step 1: Fourier Series Basics.
The Fourier series expresses a function as a sum of sine and cosine terms. The coefficients \( a_n \) (associated with the sine terms) depend on the odd powers of \( x \), while the coefficients \( b_n \) (associated with the cosine terms) depend on the even powers of \( x \).

Step 2: Analyze the given function \( f(x) = px^4 + qx^5 \).
- The term \( px^4 \) is an even function, and it contributes to the \( b_n \) (cosine) terms. Therefore, \( b_n \) depends on \( p \). - The term \( qx^5 \) is an odd function, and it contributes to the \( a_n \) (sine) terms. Therefore, \( a_n \) depends on \( q \).

Step 3: Evaluate the options.
- (A) \( a_n \) depends on \( p \): Incorrect. \( a_n \) depends on \( q \), not \( p \). - (B) \( a_n \) depends on \( q \): Correct. \( a_n \) comes from the \( qx^5 \) term. - (C) \( b_n \) depends on \( p \): Correct. \( b_n \) comes from the \( px^4 \) term. - (D) \( b_n \) depends on \( q \): Incorrect. \( b_n \) depends on \( p \), not \( q \).
Thus, the correct answer is (B) and (C): \( a_n \) depends on \( q \) and \( b_n \) depends on \( p \).
\[ \boxed{\text{The correct answers are (B) and (C).}} \]