Let `f:(0,pi) rarr R` be a twice differentiable function such that `lim_(t rarr x) (f(x)sint-f(t)sinx)/(t-x) = sin^(2) x` for all `x in (0,pi)`. If `f((pi)/(6))=(-(pi)/(12))` then which of the following statement (s) is (are) TRUE?

Options

A. `f(x(pi)/(4))=(pi)/(4sqrt(2))`
B. `f(x)lt(x^(4))/(6)-x^(2)for all x in(0,pi)`
C. There exist `alpha in (0,pi)` such that `f'(alpha)=0`
D. `f''((pi)/(2))+f((pi)/(2))`=0

Explanation

To solve the problem, we need to analyze the given limit condition and the value of the function at a specific point. Here’s a step-by-step breakdown of the solution:

### Step 1: Analyze the limit condition
We are given the limit:
\[ \lim_{t \to x} \frac{f(x) \sin t - f(t) \sin x}{t - x} = \sin^2 x \]
This limit resembles the definition of the derivative. We can interpret this as a form of the derivative of a function involving \( f(t) \) and \( \sin x \).

### Step 2: Rewrite the expression
To make the limit clearer, we can rewrite it:
\[ \lim_{t \to x} \frac{f(x) \sin t - f(t) \sin x}{t - x} = \sin^2 x \]
This suggests that we can differentiate \( f(t) \sin x \) with respect to \( t \) and evaluate it at \( t = x \).

### Step 3: Apply L'Hôpital's Rule
Since the limit is in the indeterminate form \( \frac{0}{0} \) when \( t \to x \), we can apply L'Hôpital's Rule:
\[ \frac{d}{dt}(f(x) \sin t - f(t) \sin x) = f(x) \cos t - f'(t) \sin x \]
Evaluating the derivative at \( t = x \):
\[ f(x) \cos x - f'(x) \sin x = \sin^2 x \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ f(x) \cos x - \sin^2 x = f'(x) \sin x \]
This is a first-order differential equation involving \( f(x) \).

### Step 5: Solve the differential equation
To solve for \( f(x) \), we can rearrange:
\[ f'(x) = \frac{f(x) \cos x - \sin^2 x}{\sin x} \]
This equation can be solved using an integrating factor or separation of variables.

### Step 6: Use the initial condition
We are given that \( f\left(\frac{\pi}{6}\right) = -\frac{\pi}{12} \). We can use this condition to find the specific solution for \( f(x) \).

### Step 7: Determine the statements
After solving the differential equation and applying the initial condition, we will check which of the provided statements about \( f(x) \) are true.