JEE MAIN - Mathematics (2021 - 31st August Morning Shift - No. 18)
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 ___________.
the value of |R $$-$$ S| is ___________.
Answer
2
Explanation
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
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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