Question

Let $f\left(x\right)=a{x}^3+b{x}^2+cx+1$ have extrema at $x=\alpha ,\beta$ such that $\alpha \beta <0$ and $f\left(\alpha \right)f\left(\beta \right)<0$. Then the equation $f\left(x\right)=0$ has

• A. Three equal real roots
• B. One negative root if $f\left(\alpha \right)<0$ and $f\left(\beta \right)>0$
• C. One negative root if $f\left(\alpha \right)>0$ and $f\left(\beta \right)<0$
• D. None of the above.

Answer 1

The correct option is D None of the above.

$f\left(x\right)=a{x}^3+b{x}^2+cx+1$

Extrema at $x=\alpha ,\beta$ and $\alpha \beta <0$ and $f\left(\alpha \right)f\left(\beta \right)<0$.

As $\alpha \beta <0$, there will be a root $\gamma$ between $\alpha$ and $\beta$.

Therefore, $\gamma$ is the third root and may be positive or negative.

None of the above is the correct answer.