Question

If equation $x^3+ax+b=0$ have only real roots, then

• A. ${4a}^3+27b^2<0$
• B. ${4a}^3+27b^2=0$
• C. ${4a}^3+27b^2>0$
• D. None of the above

Answer 1

Answer: (A), (B)

${4a}^3+27b^2<0$
${4a}^3+27b^2=0$

Given equation is $x^3+ax+b=0$
If the maximum lies above the real axis and the minimum lies below the real axis, then the equation will have all the roots real.
Let $f\left(x\right)=x^3+ax+b$,
then for extreme values $f^{\prime }\left(x\right)=0$
$\Rightarrow 3x^2+a=0$
$f^{\prime \prime }\left(x\right)=6x$
$\Rightarrow$ $f(-\sqrt{\frac{-a}{3}})\ge 0$ and $f\left(\sqrt{\frac{-a}{3}}\right)\le 0$
Combining the two conditions gives
$4a^3+27b^2\le 0$