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Question
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
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Solution
We have f (x) = x2 + ax + 1
= f' (x) = 2x + a
If 1 < x < 2
= 2 < 2x < 4
= 2 + a < 2x + a < 4 + a
= 2 + a < f' (x) < 4 + a
Now f (x) is strictly increasing on (1, 2) only if f' (x) > 0 for 1 < x < 2
= 2 + a ≥ 0
= a ≥ -2
∴ Required least value of a is -2
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