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The number of distinct real roots of x^4...

The number of distinct real roots of `x^4-4x^3+12 x ^2+x-1=0` is ________

Text Solution

Verified by Experts

The correct Answer is:
2
` f (x) = x ^(4) - 4 x ^(3) + 12 x ^(2) + x - 1 `
`f' (x) = 4x ^(3) - 12 x ^(2) + 24 x + 1 `
` f'' (x) = 12 x ^(2) - 24 x + 24 =12 ( x^(2) - 2x + 2 ) `
` " " = 12 { (x - 1)^(2) + 1 } gt 0 AA x `
`rArr f ' (x) ` is increasing.
Since, ` f ' (x) ` is cubic and increasing.
`rArr f ' (x) ` has only one real root and two imaginary roots.
`therefore f (x)` cannot have all distinct roots.
`rArr ` Atmost 2 real roots.
Now, ` f (-1) = 15 , f (0) = - 1 , f (1) = 9`
` therefore f (x)` must have one root in ` (-1, 0)` and othere in ` (0, 1 )`.
`rArr 2 ` real roots.
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