Question

If $x=\alpha$ and $x=\beta$ are the points of inflection $(\alpha <\beta )$ of the curve $y=3x^4-4x^3,$ then the value of $2\alpha +3\beta$ is

Answer 1

Given, $y=3x^4-4x^3$
$\Rightarrow y^{\prime }=12x^3-12x^2$
$\Rightarrow y^{\prime \prime }=36x^2-24x$
For P.O.I. $y^{\prime \prime }=0$
$\Rightarrow 36x^2-24x=0$
$\Rightarrow 12x(3x-2)=0$
$\Rightarrow x=0$ and $x=\frac{2}{3}$
So, $\alpha =0$ and $\beta =\frac{2}{3}$
$\Rightarrow 2\alpha +3\beta =2$