Question

If the equation $f (x) =x^{3}+3x^{2}-9x+a=0 \,\forall\,a \le\,R$ has one distinct & two identical roots, then the total number of values of a equals

• A. $0$
• B. $1$
• C. $3$
• D. $2$

Answer 1

Answer: (D)

Given $f (x) =x^{3}+3x^{2}-9x+a=0 \, \dots(i)$
$\therefore f'(x) =3x^{2}+6xc-9=3[(x^{2}+2x-3)]$,
here $a>\,0, c <\,0$
$\therefore D>\,0$
$\Rightarrow f'(x) = \alpha$ has two distinct roots $\alpha, \beta$ where $(\alpha=-3, \beta=1)$, therefore $f (\alpha) f (\beta)=0$ gives the values of a.
Now $f(-3) f (1)=0$
$\Rightarrow (a-5)(a+27)=0$
$\therefore a=5$ or $27$
$\therefore $ Number of values $= 2$