JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Application of Derivatives

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 ______.

Answer: 2

Solution

For

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

, we have

f^{'} (x) = 2 x + a

. For

f

to be increasing on

[1, 2]

, we need

f^{'} (x) \geq 0

for all

x \in [1, 2]

. The minimum of

f^{'} (x)

[1, 2]

is at

x = 1

2 (1) + a \geq 0

, so

a \geq - 2

. Thus

R = - 2

. For

f

to be decreasing on

[1, 2]

, we need

f^{'} (x) \leq 0

for all

x \in [1, 2]

. The maximum of

f^{'} (x)

[1, 2]

is at

x = 2

2 (2) + a \leq 0

, so

a \leq - 4

. Thus

S = - 4

. Therefore,

∣ R - S ∣ = ∣ - 2 - (- 4) ∣ = 2

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Integer answer