Verified Solution • 2025–26 Session moderate

QJEEM21Mabca0: JEE Main 2021 Mathematics - Class 12 JEEM21Mabca0 Solution: If 'R' is the least value of 'a' such that the function f(x) =…

If 'R' is the least value of 'a' such that the function f(x) = x2 + ax + 1 is increasing on [1, 2] and 'S' is the greatest value of 'a' such that the function f(x) = x2 + ax + 1 is decreasing on [1, 2], then
the value of |R −- S| is ___________.

Step-by-Step Verified Solution

f(x) = x2 + ax + 1

f'(x) = 2x + a

when f(x) is increasing on [1, 2]

2x + a ≥\ge 0 ∀\forall x ∈\in [1, 2]

a ≥\ge −- 2x ∀\forall x ∈\in [1, 2]

R = −- 4

when f(x) is decreasing on [1, 2]

2x + a ≤\le 0 ∀\forall x ∈\in [1, 2]

a ≤\le −- 2 ∀\forall x ∈\in [1, 2]

S = −- 2

|R −- S| = | −- 4 + 2 | = 2

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