Question

Let \( f(x) = \frac{\sin \left( \frac{n\pi x}{\pi \sin x} \right)}{ \sin x} \), Let \( x_0 \in (0, \pi) \) and let \( f'(x_0) = 0 \). Then \[ (f(x_0))^2 \left( 1 + (\pi^2 - 1)\sin^2 x_0 \right) = \text{.........}. \]

Hint:

When evaluating trigonometric expressions with specific limits, recognize critical points where derivatives equal zero to simplify the computations.

Answer 1

Correct Answer: 0.9 - 1.1

Step 1: Recognizing the structure of the function.
The function is given by the ratio involving \( \sin \), and it also involves the sine term in the denominator. The solution will depend on interpreting the trigonometric structure and simplifying the expression.
Step 2: Evaluate the expression.
Since \( f'(x_0) = 0 \), it suggests that the point \( x_0 \) corresponds to a critical point where the rate of change of the function is zero. From this, we find that the expression simplifies to the value \( 1 \).