Question

If the polynomial $2x^{3}-3\left(\right.a+1\left.\right)x^{2}+6ax-12$ 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

Let $f\left(x\right)=2x^{3}-3\left(a + 1\right)x^{2}+6ax-12$
For maxima-minima, $f^{'}\left(x\right)=6\left\{x^{2} - \left(a + 1\right) x + a\right\}=0$
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
$\therefore a=2$