Question

Let \( M \) and \( m \) respectively be the maximum and the minimum values of \[ f(x) = \frac{1 + \sin^2 x}{\cos^2 x} + \frac{4 \sin 4x}{\sin^2 x \cos^2 x} \quad {for} \quad x \in \mathbb{R} \] Then \( M^4 - m^4 \) is equal to:

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

Hint:

When solving optimization problems involving trigonometric functions, use their periodicity and symmetry to simplify calculations and identify maximum and minimum values.

Answer 1

The Correct Option is D

Step 1: Begin by analyzing the function \( f(x) \), and separate the terms to find the maximum and minimum values of the given expression. Use the periodicity of trigonometric functions to simplify and solve. 
Step 2: To find the maximum and minimum values, take the derivative of \( f(x) \) and solve for the critical points. Analyze the behavior of \( f(x) \) at these points and the boundaries of the domain. 
Step 3: After finding the maximum and minimum values \( M \) and \( m \), calculate \( M^4 - m^4 \). Thus, the correct answer is (4).