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Question

Let f(x)=ax3+bx2+cx+1 have extrema at x=α,β such that αβ<0 and f(α)f(β)<0. Then the equation f(x)=0 has

A
Three equal real roots
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B
One negative root if f(α)<0 and f(β)>0
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C
One negative root if f(α)>0 and f(β)<0
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D
None of the above.
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Solution

The correct option is D None of the above.
f(x)=ax3+bx2+cx+1
Extrema at x=α,β and αβ<0 and f(α)f(β)<0.
As αβ<0, there will be a root γ between α and β.
Therefore, γ is the third root and may be positive or negative.
None of the above is the correct answer.

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