Question

Let a cubic polynomial be $f(x)=\frac{x^3}{3}-x^2+a x+2$. If all the values of a for which $f(x)$ has a positive point of maximum also satisfy the inequality $3 x^2-(b+1) x+b(b-2)<0$ and value of $b$ lies in $[p, q]$, then find the value of $(p+q)$.

Answer 1

For a positive point of maximum both distinct roots of $f^{\prime}(x)=0$ should be greater than zero.
$\therefore f^{\prime}(x)=x^2-2 x+a$
$\Theta \text { both distinct greater than zero }$
(i) $D >0 \Rightarrow 4-4 a >0 \Rightarrow a <1$
(ii) $f ^{\prime}(0)>0 \Rightarrow a >0$
(iii)$\frac{-(-1)}{2 \cdot \frac{1}{3}}=\frac{3}{2}>0 $
$\therefore a \in(0,1)$
$\Theta a \in(0,1) \text { satisfying } 3 x ^2-( b +1) x + b ( b -2)<0$
$\therefore b ( b -2) \leq 0 \Rightarrow 0 \leq b \leq 2$
$\text { and } 3-( b +1)+ b ( b -2) \leq 0 $
$\Rightarrow b ^2-3 b +2 \leq 0 \Rightarrow 1 \leq b \leq 2 $
$\therefore b \in[1,2] $
$\therefore p + q =3 $