Question

If the polynomial $2x^3-3(a+1)x^2+6ax-12$ has a local maximum at $x_1$ and a local minimum at $x_2$ and if $2x_1=x_2$ , then the value of $2a$ such that $a<1$ is

Answer 1

Answer: 1

Let $f\left(x\right)=2x^3-3\left(a+1\right)x^2+6ax-12$
For maxima-minima, $f^{{}^{\prime }}\left(x\right)=6\left\{x^2-\left(a+1\right)x+a\right\}=0$
i.e. $6\left(x-1\right)\left(x-a\right)=0\Rightarrow x=1,a$
$f^{"}\left(x\right)=12x-6\left(a+1\right)$
$f^{"}\left(a\right)=12a-6a-6=6a-6$
$=6\left(a-1\right)$
$\therefore$ If $a<1$ , then $x=a$ is a local maxima and $x=1$ is a local minima
$\therefore 2a=1$
If $a>1$ , then $x=a$ is a local minima and $x=1$ is a local maxima
$\therefore a=2$