Let f : 0 , π → R be a twice differentiable function such that lim t → x f x sin t - f t sin x t - x = sin 2 x for all x ∈ 0 , π . If f π 6 = - π 12 , then which of the following statement(s) is (are) T

Question

Let f : 0 , &#960; &#8594; R be a twice differentiable function such that lim t &#8594; x f x sin t - f t sin x t - x = sin 2 x for all x &#8712; 0 , &#960; . If f &#960; 6 = - &#960; 12 , then which of the following statement(s) is (are) TRUE ?

Quantrex Academy · Accepted Answer

Correct answer: B. f x &#60; x 4 6 - x 2 for all x &#8712; 0 , &#960;

lim t → x f x sin t - f t sin x t - x = sin 2 x By using L'Hopital rule lim t → x f x cos t - f ′ t sin x 1 = sin 2 x ⇒ f x cos x - f ′ x sin x = sin 2 x ⇒ - f ′ x sin x - f x cos x sin 2 x = 1 ⇒ - d f x sin x d x = 1 ⇒ f x sin x = - x + c Put x = π 6 and f π 6 = - π 12 ∴ c = 0 ⇒ f x = - x sin x A f π 4 = - π 4 1 2 B f x = - x sin x As sin x > x - x 3 6 , - x sin x - x 2 + x 4 6 ∴ f x - x 2 + x 4 6 ∀ x ∈ 0 , π C f ′ x = - sin x - x cos x f ′ x = 0 ⇒ tan x = - x ⇒ there exist α ∈ 0 , π for which f ′ α = 0 D f " x =

Let f : 0 , π → R be a twice differentiable function such that lim t → x f x sin t - f t sin x t - x = sin 2 x for all x ∈ 0 , π . If f π 6 = - π 12 , then which of the following statement(s) is (are) TRUE ?

Options

Correct answer

Step-by-step solution

More from Differential Equations

Let f : 0 , &#960; &#8594; R be a twice differentiable function such that lim t &#8594; x f x sin t - f t sin x t - x = sin 2 x for all x &#8712; 0 , &#960; . If f &#960; 6 = - &#960; 12 , then which of the following statement(s) is (are) TRUE ?

Options

Correct answer

Step-by-step solution

More from Differential Equations

Let f : 0 , π → R be a twice differentiable function such that lim t → x f x sin t - f t sin x t - x = sin 2 x for all x ∈ 0 , π . If f π 6 = - π 12 , then which of the following statement(s) is (are) TRUE ?