Question

Find the intervals of increase and decrease of the following function.
$y=-x(x-2{)}^2$

Answer 1

Given $y=-x(x-2{)}^2$

Calcullating $\frac{dy}{dx}$ for intervals of increasing and decreasing functions.

$\frac{dy}{dx}=-\left[x\frac{d}{dx}\right(x-2{)}^2+\frac{d}{dx}\left(x\right)\cdot (x-2{)}^2]$

$=-[x\cdot 2(x-2)+(x-2{)}^2]$

$=-(x-2)[2x+x-2]$

$=-(x-2)[3x-2]$

increasing function $\frac{dy}{dx}>0$

$\Rightarrow -(x-2)(3x-2)>0$

$\Rightarrow (x-2)(3x-2)<0$

$x\in (\frac{2}{3},2)$

and for decreasing

$\frac{dy}{dx}<0$

$\Rightarrow -(x-2)(3x-2)<0$

$(x-2)(3x-2)>0x\in (-\infty ,\frac{2}{3})$ and $(2,\infty )$.