Question

Find the intervals in which the function $f\left(x\right)=(x-1)(x-2{)}^2$ is increasing or decreasing.

Answer 1

$f\left(x\right)=(x-1)(x-2{)}^2$

$f^{\prime }\left(x\right)=(x-2{)}^2+2(x-1\left)\right(x-2)$

$f^{\prime }\left(x\right)=(x-2)(x-2+2x-2)$

$f^{\prime }\left(x\right)=(x-2)(3x-4)$

$f^{\prime }\left(x\right)>0$ for $x>2$ or $x<\frac{4}{3}$ it is increasing

$f^{\prime }\left(x\right)<0$ for $x<2$ and $x>\frac{4}{3}$ it is decreasing