Multiple choice

Let $f(x)=ax^3+bx^2+cx+1$ have extrema at $x=\alpha,\beta$ such that $\alpha \beta<0$ and $f(\alpha)f(\beta)<0$. Then the equation $f(x)=0$ has

  1. Three equal real roots
  2. One negative root if $f(\alpha)<0$ and $f(\beta)>0$
  3. One negative root if $f(\alpha)>0$ and $f(\beta)<0$

  4. None of the above.
Reveal answer Fill a bubble to check yourself
D Correct answer
AI explanation

The given conditions state that the extrema alpha and beta have opposite signs and their corresponding function values also have opposite signs. Since the function values at the local maximum and local minimum have opposite signs, the cubic graph crosses the x-axis exactly once between the extrema and once outside, meaning the equation f(x) = 0 has three distinct real roots. This specific outcome is not described by any of the previous statements, so none of the above is the correct choice.

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