If α is the only real root of x 3 + b x 2 + c x + 1 = 0 b c , then the value of α is (where, . represents the greatest integer function)

Question

If α is the only real root of x 3 + b x 2 + c x + 1 = 0 b c , then the value of α is (where, . represents the greatest integer function),

MARKS App · Accepted Answer

Let, f x = x 3 + b x 2 +c x + 1 f x = 0 has only one real root so it is an increasing function Now, f 0 = 1 , f - 1 = - 1 + b -c + 1 f - 1 = b -c 0 using mean value theorem, real root must lie between - 1,0 so, α = - 1 α = 1

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