Question

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

Answer 1

Answer: 3

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