Question

Let \( M \) and \( m \) respectively be the maximum and the minimum values of \( f(x) = \begin{vmatrix} 1 + \sin^2x & \cos^2x & 4\sin4x \\ \sin^2x & 1 + \cos^2x & 4\sin4x \\ \sin^2x & \cos^2x & 1 + 4\sin4x \end{vmatrix}, \quad x \in \mathbb{R} \)  for \( x \in \mathbb{R} \). Then \( M^4 - m^4 \) is equal to:

• A. 1215
• B. 1040
• C. 1295
• D. 1280

Hint:

For functions involving trigonometric expressions, use trigonometric identities and calculus techniques to find extrema.

Answer 1

The Correct Option is D

This task involves analyzing the determinant of a specified matrix and then finding the maximum and minimum values of the resultant function, denoted as \( f(x) \). The process is delineated as follows:

1. Matrix Specification:
The matrix under consideration is:

\( \begin{vmatrix} 1 + \sin^2x & \cos^2x & 4\sin4x \\ \sin^2x & 1 + \cos^2x & 4\sin4x \\ \sin^2x & \cos^2x & 1 + 4\sin4x \end{vmatrix}, \quad x \in \mathbb{R}\)

2. Determinant Simplification via Row Operations:
To facilitate the calculation of the determinant, the following row operations are applied:

\( R_2 \to R_2 - R_1 \) and \( R_3 \to R_3 - R_1 \):

\( \begin{vmatrix} 1 + \sin^2x & \cos^2x & 4\sin4x \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{vmatrix}\)

3. Determinant Expansion and Simplification:
The determinant is expanded along the first row, yielding \( f(x) \):

\( f(x) = (1 + \sin^2x) \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} - \cos^2x \begin{vmatrix} -1 & 0 \\ -1 & 1 \end{vmatrix} + 4\sin4x \begin{vmatrix} -1 & 1 \\ -1 & 0 \end{vmatrix}\)

The minor determinants are evaluated as:

\( \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} = 1, \quad \begin{vmatrix} -1 & 0 \\ -1 & 1 \end{vmatrix} = -1, \quad \begin{vmatrix} -1 & 1 \\ -1 & 0 \end{vmatrix} = 1\)

Substituting these values back into the expansion results in:

\( f(x) = (1 + \sin^2x)(1) - \cos^2x(-1) + 4\sin4x(1)\)

Further simplification yields:

\( f(x) = 1 + \sin^2x + \cos^2x + 4\sin4x\)

Applying the Pythagorean identity \( \sin^2x + \cos^2x = 1 \), the function simplifies to:

\( f(x) = 1 + 1 + 4\sin4x = 2 + 4\sin4x\)

4. Determination of Maximum and Minimum Values:
The function \( f(x) = 2 + 4\sin4x \) is dependent on \( \sin4x \), which ranges between \(-1\) and \(1\).

The maximum value of \( f(x) \) is achieved when \( \sin4x = 1 \):

\( \text{Maximum value of } f(x): \quad f(x) = 2 + 4(1) = 6\)

The minimum value of \( f(x) \) is achieved when \( \sin4x = -1 \):

\( \text{Minimum value of } f(x): \quad f(x) = 2 + 4(-1) = -2\)

Therefore, the maximum value \( M = 6 \) and the minimum value \( m = -2 \).

5. Calculation of \( M^4 - m^4 \):
Using the determined values of \( M \) and \( m \):

\( M^4 - m^4 = 6^4 - (-2)^4\)

The individual terms are calculated as:

\( 6^4 = 1296, \quad (-2)^4 = 16\)

The final subtraction yields:

\( M^4 - m^4 = 1296 - 16 = 1280\)

Conclusion:
The computed value of \( M^4 - m^4 \) is \( \boxed{1280} \).