Question

Find the value(s) of $x$ for which $y=\left[x\right(x-2{)}^2]$ is an increasing function ?

Answer 1

$y=\left[x\right(x-2){]}^2$

$\frac{dy}{dx}=2\left[x\right(x-2\left)\right][x.1+(x-2\left).1\right]$

$=2x(x-2)(2x-2)$

$=4(x-0)(x-1)(x-2)$

Now,$\frac{dy}{dx}>0if,xϵ(0,1)\cup (2,\infty ),\frac{dy}{dx}<0,$

if,$xϵ(-\infty ,0)\cup (1,2)$

Hence,for $xϵ(0,1)\cup (2,\infty )$, given function is increasing.