Let f x = e x - x and g x = x 2 - x , ∀ x ϵ R . Then the set of all x ϵ R , where the function h x = f o g x is increasing, is:

Question

Let f x = e x - x and g x = x 2 - x , ∀ x ϵ R . Then the set of all x ϵ R , where the function h x = f o g x is increasing, is:, - 1 , - 1 2 ⋃ 1 2 , ∞, [ 0 , ∞ ), 0 , 1 2 ∪ [ 1 , ∞ ), - 1 2 , 0 ∪ [ 1 , ∞ )

MARKS App · Accepted Answer

h x = f o g x = e g x - g x = e x 2 - x - x 2 - x h ' x = e x 2 - x × 2 x - 1 - 2 x + 1 = 2 x - 1 e x 2 - x - 1 ∵ h x is increasing ∴ h ' x ≥ 0 ⇒ 2 x - 1 e x 2 - x - 1 ≥ 0 Case 1: 2 x - 1 ≥ 0 & x 2 - x ≥ 0 ⇒ x ≥ 1 2 & x ∈ - ∞ , 0 ∪ 1 , ∞ ⇒ x ∈ 1 , ∞ Case 2: 2 x - 1 ≤ 0 and x 2 - x ≤ 0 x 1 2 and x ∈ 0 , 1 ⇒ x ∈ 0 , 1 2 ∴ x ∈ 0 , 1 2 ∪ [ 1 , ∞ )

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