If R is the least value of a such that the function f x = x 2 + a x + 1 is increasing on [ 1 , 2 ] and S is the greatest value of a such that the function f ( x ) = x 2 + a x + 1 is decreasing on [ 1 , 2 ] , then the value of | R - S | is

Step-by-step solution

If f ' ( x ) = 2 x + a ≥ 0 ∀ x ∈ [ 1 , 2 ]    ⇒ a ≥ - 2 x ∀ x ∈ [ 1 , 2 ] a min = R = - 2 If a m a x = S = - 4 | R - S | = 2 Alternate : If f ( x ) is increasing for all x ∈ [ 1 , 2 ] then - a 2 ≤ 1 ⇒ a ≥ - 2 ⇒ R = - 2 If f ( x ) is decreasing for x ∈ [ 1 , 2 ] , then - a 2 ≥ 2 ⇒ a ≤ - 4 ⇒ S = - 4 | R - S | = 2