Let f ⁡  :  1 2 1 ⟶ R (the set of all real numbers) be a positive, non-constant and differentiable function such that f ' x < 2 f ⁡ x and f ⁡ 1 2 = 1 . Then the value of ∫ 1 / 2 1 f ⁡ x dx lies in the interval:

Question

Let f &#8289; &#160;:&#160; 1 2 1 &#10230; R (the set of all real numbers) be a positive, non-constant and differentiable function such that f ' x &#60; 2 f &#8289; x and f &#8289; 1 2 = 1 . Then the value of &#8747; 1 / 2 1 f &#8289; x dx lies in the interval:

Aicharya · Accepted Answer

f ' x &#60; 2 f x &#8658; f ' x f x &#60; 2 &#8658; &#8747; 1 2 x f ' x f x d x = &#8747; 1 2 x 2 d x &#8658; ln f x &#60; 2 x - 1 &#8658; f x &#60; e 2 x - 1 &#8658; 0 &#60; &#8747; 1 2 1 f x d x &#60; &#8747; 1 2 1 e 2 x - 1 d x &#8658; 0 &#60; &#8747; 1 2 1 f x d x &#60; e 2 x - 1 2 1 2 1 &#8658; 0 &#60; &#8747; 1 2 1 f x d x &#60; e - 1 2

Let f ⁡  :  1 2 1 ⟶ R (the set of all real numbers) be a positive, non-constant and…

Solution

Let f &#8289; &#160;:&#160; 1 2 1 &#10230; R (the set of all real numbers) be a positive, non-constant and…

Solution

Let f ⁡ : 1 2 1 ⟶ R (the set of all real numbers) be a positive, non-constant and…