JEE Main 202131 AugustMorning Shift

Exam: JEE Main
Year: 2021
Date: 31 August
Shift: Morning Shift
Subject: Mathematics
Chapter: Application of Derivatives

JEE Main 2021 · 31 August · Morning Shift · Mathematics · Application of Derivatives

If R is the least value of a such that the function f(x) = x ^ 2 +...

R

is the least value of

a

such that the function

f\left(x\right) = {x}^{2} + ax + 1

is increasing on

[ 1, 2 ]

and

S

is the greatest value of

a

such that the function

f ( x ) = {x}^{2} + ax + 1

is decreasing on

[ 1, 2 ]

, then the value of

| R - S |

If R is the least value of a such that the function f(x) = x ^ 2 + ax + 1 is increasing on [ 1, 2 ] and S is the

Correct answer

Step-by-step solution

If ${f}^{'} ( x ) = 2x + a \ge 0 \forall x \in [ 1, 2 ] ⇒ a \ge - 2x \forall x \in [ 1, 2 ]$

${a}_\min = R = - 2$ If

${a}_\max = S = - 4$

$| R - S | = 2$

Alternate :

If $f ( x )$ is increasing for all $x \in [ 1, 2 ]$ then

$- \dfrac{a}{2} \le 1 ⇒ a \ge - 2 ⇒ R = - 2$

If $f ( x )$ is decreasing for $x \in [ 1, 2 ] ,$ then

$- \dfrac{a}{2} \ge 2 ⇒ a \le - 4 ⇒ S = - 4$

If R is the least value of a such that the function f(x) = x ^ 2 + ax + 1 is increasing on [ 1, 2 ] and S is the

$| R - S | = 2$

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