Question

The curve \( y = \frac{x}{1 + x \tan{x}} \) attains maxima at:

• A. at \( x = \cos{x} \)
• B. at \( x = -\cos{x} \)
• C. at \( x = \cos{x} \) and for some \( x \in \left[ -2\pi, \pi \right] \)
• D. for some \( x \in \left[ -2\pi, \pi \right] \)

Hint:

For maximum or minimum points of a curve, differentiate the function and set the derivative equal to zero to find critical points.

Answer 1

The Correct Option is A

Step 1: Consider the function and find the derivative.
The given curve is: \[ y = \frac{x}{1 + x \tan{x}}. \] To find the points where maxima occur, we take the derivative of \( y \) with respect to \( x \) and solve for \( x \). Setting the derivative equal to zero will help us find the critical points.

Step 2: Differentiate the function.
The derivative \( \frac{dy}{dx} \) involves applying the quotient rule and simplifying. After differentiating and simplifying, we find that the maxima occur at \( x = \cos{x} \).

Step 3: Conclusion.
Thus, the correct answer is (a) at \( x = \cos{x} \).