Question

Let f(x) = eˣ – x and g(x) = x² – x, ∀ x ∈ R. Then the set of all x ∈ R, where the function h(x) = (fog) (x) is increasing is: (A) [0, 1/2] ∪ [1, [infinity]]
(B) [1, 1/2] ∪ [1/2, [infinity]]
(C) [-1/2, 0] ∪ [1, [infinity]]
(D) [0, [infinity]]

Answer 1

answer : option (A) [0, 1/2 ] U [1 , ∞)

given, f(x) = eˣ – x and g(x) = x² – x, ∀ x ∈ R. Then the set of all x ∈ R, where the function h(x) = (fog) (x).

h(x) = f(g(x))

= f(x² - x)

= e^(x² - x) - (x² - x)

= e^(x² - x) - x² + x

differentiating with respect to x,

h'(x) = e^(x² - x)(2x - 1) - 2x + 1

= (2x - 1)[e^(x² - x) - 1]

for h'(x) = 0 ⇒x = 1/2 and e^(x² - 1) = 1

e^(x² - x) = e^0 ⇒x² - x = 0

⇒x = 0, 1

we know, function h(x) is increasing when h'(x) > 0

putting 0, 1/2 and 1 in number line as shown in figure. and get value of positive part

I.e., function is increasing when x ≥ 1 or 0 ≤ x ≤ 1/2

option (A) is correct choice.

Answer 2

Answer:

15 points......................