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If alpha is the root (having the least a...

If `alpha` is the root (having the least absolute value) of the equation `x^2-b x-1=0(b in R^+)` , then prove that `-1ltalphalt0.`

Text Solution

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Let `f(x) = x^(2) - bx - 1 (b in R^(+))`
`f( -1) = b + ve`
`f(0) = - 1= - ve`
`f(1) = - b = -ve `
Clearly, one root lies in `(-1,0)` and the other in `(1,infty)` . Now , sum
of roots is b, which is positive . So .`alpha ` (having the least absolute
value) `in (-1, 0)` .
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