Question

Let $f\left(x\right)=x^3+a{x}^2+bx+c=0$ be the cubic equation, where $a,b,c\in R.$ Now, $f^{\prime }\left(x\right)=3x^2+2ax+b.$
Let $D$ be the discriminant of the equation $f^{\prime }\left(x\right)=0$.
If $D>0$ and $f\left(x_1\right)\cdot f\left(x_2\right)<0$, where $x_1$ and $x_2$ are the roots of $f^{\prime }\left(x\right)=0,$ then

• A. $f\left(x\right)=0$ has all real and distinct roots
• B. $f\left(x\right)=0$ has three real roots but one of the roots would be repeated
• C. $f\left(x\right)=0$ would have just one real root
• D. None of the above

Answer 1

The correct option is A $f\left(x\right)=0$ has all real and distinct roots

Here $f\left(x\right)=0$ will have three distinct root $\alpha ,\beta ,\gamma$
From graph $f\left(x_1\right)\cdot f\left(x_2\right)<0$
$\therefore f\left(x\right)$ has real and distinct roots.