Question

$f(x)=x^{3}-9 x^{2}+24 x+c=0$ has three real and distinct roots, $\alpha, \beta$, and $\gamma$
If $[\alpha]+[\beta]+[\gamma]=7$, then the values of $c$, where [.] represents the greatest integer function, are

• A. (-20, - 16)
• B. (-20, -18)
• C. (-18, -16)
• D. none of these

Answer 1

Answer: (B)

Let $g(x)=x^{3}-9 x^{2}+24 x=x\left(x^{2}-9 x+24\right)$
$\Rightarrow g'(x)=3(x-2)(x-4)$
Sign scheme of $g(x)$

For three real roots of
$f ( x )= x ^{3}-9 x ^{2}+24 x + c =0, c$ must lie in the interval $(-20,-16)$
$f (0)= c <0$
$f (1)=1-9+24+ c = c +16<0$
for $\forall x \in(-20,-16)$
$f (2)=8-36+48+ c = c +20>0$
$\alpha \in(1,2)$
$f (3)=27-81+72+ c =18+ c$
$\Rightarrow f (3)<0$
if $c \in(-20,-18)$
or $f (3)>0$
if $c \in(-18,-16)$
and $\beta \in(3,4)$ if $c \in(-18,-16)$
Now $f(4)=64-144+96+c$
$=16+c< 0 \forall c \in(-20,-16)$
$f(5)=125-225+120+c$
$=c+20>0 \forall c \in(-20,-16)$
$\Rightarrow \gamma \in(4,5) $
$ \Rightarrow [\gamma]=4$
Thus, $[\alpha]+[\beta]+[\gamma]=\begin{cases}1+2+4, & -20 < c < -18 \\ 1+3+4, & -18< c < -16\end{cases}$
Now if $c \in(-20,-18)$
$\alpha \in(1,2), \beta \in(2,3), \gamma \in(4,5)$
$\Rightarrow [\alpha]+[\beta]+[\gamma]=7$
if $c \in(-18,-16), \alpha \in(1,2), \beta \in(3,4), \gamma \in(4,5)$
$\Rightarrow [\alpha]+[\beta]+[\gamma]=8$