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Question
Choose the correct option from the given alternatives :
If x = –1 and x = 2 are the extreme points of y = αlogx + βx2 + x`, then ______.
Options
α = –6, β = `1/2`
α = –6, β = `-1/2`
α = 2, β = `-1/2`
α = 2, β = `1/2`
Solution
If x = –1 and x = 2 are the extreme points of y = αlogx + βx2 + x, then `bb(underline(α = 2, β = -1/2))`
Explanation:
[Hint: y = αlogx + βx2 + x
∴ `dy/dx = α/x + β xx 2x + 1`
= `α/x + 2βx + 1`
f(x) has extreme values at x = –1 and x = 2
∴ f'(–1) = 0 and f(2) = 0
∴ `α + 2β` = 1
and `α/(2) + 4β` = –1
By solving these two equations, we get
α = 2, β = `-(1)/(2)`].
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