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 , &#160; &#8704; &#160; x &#160; &#1013; &#160; R . Then the set of all x &#160; &#1013; &#160; R , where the function h x = f o g x is increasing, is:

Aicharya · 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 &#215; 2 x - 1 - 2 x + 1 = 2 x - 1 e x 2 - x - 1 &#8757; h x is increasing &#8756; h ' x &#8805; 0 &#8658; 2 x - 1 e x 2 - x - 1 &#8805; 0 Case 1: 2 x - 1 &#8805; 0 & x 2 - x &#8805; 0 &#8658; x &#8805; 1 2 &#160; &#38; &#160; x &#8712; - &#8734; , &#160; 0 &#8746; 1 , &#8734; &#8658; x &#8712; 1 , &#160; &#8734; Case 2: 2 x - 1 &#8804; 0 and x 2 - x &#8804; 0 x &#60; 1 2 and x &#8712; 0 , &#160; 1 &#8658; x &#8712; 0 , 1 2 &#8756; &#160; &#160; x &#8712; 0 , 1 2 &#8746; [ 1 , &#8734; )

Let f x = e x - x and g x = x 2 - x ,   ∀   x   ϵ   R . Then the set of all…

Solution

Let f x = e x - x and g x = x 2 - x , &#160; &#8704; &#160; x &#160; &#1013; &#160; R . Then the set of all…

Solution

Let f x = e x - x and g x = x 2 - x , ∀ x ϵ R . Then the set of all…