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:
Show Hint
- 1215
- 1040
- 1295
- 1280
The Correct Option is D
Solution and Explanation
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).
Top Questions on Calculus
- Prove that \(\int_0^\pi \sqrt{\frac{1+\cos 2x}{2}} \, dx = 2\).
- At \( t = 2 \), the slope of the vector function \( \vec{f}(t) = 2\hat{i} + 3\hat{j} + 5t^2\hat{k} \) is
- If function \( f \) is defined as \[ f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} \] then prove that \( f \) is continuous.
- The vector function is given by \(\vec{f}(t) = t\hat{i} + t^2\hat{j} + 5\hat{k}\), then at point \(t = 1\) the slope is
- If \(\int x \log x \, dx = \frac{x^2}{2} f(x) - \frac{x^2}{4} + c\), then \( f(x) \) is
Questions Asked in JEE Main exam
- The sum of all rational numbers in \( (2 + \sqrt{3})^8 \) is:
- JEE Main - 2025
- Binomial theorem
- The maximum covalency of a non-metallic group 15 element 'E' with the weakest E-E bond is:
- JEE Main - 2025
- Group 15 Elements
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to:
- JEE Main - 2025
- Conic sections
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is:
- JEE Main - 2025
- Number Systems
- For an experimental expression \( y = \frac{32.3 \times 1125}{27.4} \), where all the digits are significant. Then to report the value of \( y \), we should write:
- JEE Main - 2025
- Significant figures