Question

The range of values of $\alpha$ so that all the roots of the equation $2x^3-3x^2-12x+\alpha =0$ are real distinct belongs to

• A. $(7,20)$
• B. $(-7,20)$
• C. $(-20,7)$
• D. $(-7,7)$

Answer 1

The correct option is B $(-7,20)$

Let $f\left(x\right)=2x^3-3x^2-12x+\alpha$, then

$f^{\prime }\left(x\right)=6(x^2-x-2)=6(x+1)(x-2)$

So, the roots of $f^{\prime }\left(x\right)=0$ are $x=-1,2$;

Now, $f\left(x\right)=0$ will have all real roots, if $f(-1)>0$ and $f\left(2\right)<0$.

$\Rightarrow -2-3+12+\alpha >0$ and $16-12-24+\alpha <0$

$\Rightarrow -7<\alpha <20$