Let f : R → R a n d g : R → R be two non-constant differentiable functions. If f ′ x = e f x - g x g ′ x f o r a l l x ∈ R , and f 1 = g 2 = 1 then which of the following statement(s) is (are) TRUE?

Question

Let f : R &#8594; R a n d g : R &#8594; R be two non-constant differentiable functions. If f &#8242; x = e f x - g x g &#8242; x f o r a l l x &#8712; R , and f 1 = g 2 = 1 then which of the following statement(s) is (are) TRUE?

Quantrex Academy · Accepted Answer

Correct answer: B. f 2 > 1 - log e ⁡ 2 f ′ x = e f x - g x g ′ x ∀ x ∈ R ⇒ e - f x . f ′ x - e - g x g ′ x = 0 ⇒ ∫ e - f x f ′ x - e - g x . g ′ x d x = C ⇒ - e - f x + e - g x = C ⇒ - e - f 1 + e - g 1 = - e - f 2 + e - g 2 ⇒ - 1 e + e g 1 = - e - f 2 + 1 e ⇒ e - f 2 + e - g 1 = 2 e ∴ e - f 2 < 2 e a n d e - g 1 < 2 e ⇒ - f 2 < ln ⁡ 2 - 1 a n d - g 1 < ln ⁡ 2 - 1 ⇒ f 2 > 1 - ln ⁡ 2 a n d g 1 > 1 - ln ⁡ 2

Let f : R → R a n d g : R → R be two non-constant differentiable functions. If f ′ x = e f x - g x g ′ x f o r a l l x ∈ R , and f 1 = g 2 = 1 then which of the following statement(s) is (are) TRUE?

Options

Correct answer

Step-by-step solution

More from Definite Integration

Let f : R &#8594; R a n d g : R &#8594; R be two non-constant differentiable functions. If f &#8242; x = e f x - g x g &#8242; x f o r a l l x &#8712; R , and f 1 = g 2 = 1 then which of the following statement(s) is (are) TRUE?

Options

Correct answer

Step-by-step solution

More from Definite Integration

Let f : R → R a n d g : R → R be two non-constant differentiable functions. If f ′ x = e f x - g x g ′ x f o r a l l x ∈ R , and f 1 = g 2 = 1 then which of the following statement(s) is (are) TRUE?