Question

Let f:(0, π) →R be a twice differentiable function such that lim_(t →x) (f(x)sint - f(t)sinx)/(t - x) = sin²x for all x  ∈ (0, π). If f(π/6) = - π/12, then which of the following statement(s) is(are) TRUE?

(A) f(π/4) = π/4√2

(B) f(x) < (x4/6)x² for all x ∈ (0, π)

(C) There exists α ∈ (0, π) such that f'(α) = 0

(D) f''(π/2) + f(π/2) = 0

Answer 1

Answer is (B), (C), (D)

Given: f: (0, π)  → R is twice differentiable function such that

Also, it is given that

This equation can be written in differential form:

-d/dx(f(x)/sinx) = 1

 

Thus, there exists α ∈ (0, π) for which f'(x) = 0.

As f(x) is continuous in [0, π] and differentiable in (0, π). Hence, option (C) is true

Hence, option (D) is true.