Question

If \( y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots \infty}}} \), then the value of \( \frac{d^2y}{dx^2} \) at \( (\pi,1) \) is:

• A. \( 2 \)
• B. \( -2 \)
• C. \( -\frac{1}{2} \)
• D. \( \frac{1}{2} \)

Hint:

For nested functions, express them as a self-contained equation and use implicit differentiation.

Answer 1

The Correct Option is B

Step 1: Establishing the Functional Equation Since the function follows a repeating pattern, we define: \[ y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots}}} \] Since the same function appears infinitely, we square both sides: \[ y^2 = \sin x + y. \] Rearrange: \[ y^2 - y - \sin x = 0. \] 
Step 2: Differentiate Implicitly Differentiating both sides with respect to \( x \): \[ 2y \frac{dy}{dx} = \cos x + \frac{dy}{dx}. \] Rearranging: \[ \frac{dy}{dx} (2y - 1) = \cos x. \] \[ \frac{dy}{dx} = \frac{\cos x}{2y - 1}. \] 
Step 3: Compute Second Derivative Differentiating again and substituting \( x = \pi \), \( y = 1 \), we obtain: \[ \frac{d^2y}{dx^2} = -2. \] Thus, the correct answer is: \(Option (2): -2.\)