Question

Is `f(x)=(x-2)^2(x+1)` increasing or decreasing at `x=1`?

Answer 1

`f(x)" is decreasing at "x=1`

Explanation:

`"to determine if f(x) is increasing/decreasing at x = 1"`

`"differentiate f(x) and evaluate at x = 1"`

`• " if "f'(1)>0" then "f(x)" is increasing at x = 1"`

`• " if "f'(1)<0" then "f(x)" is decreasing at x = 1"`

`f(x)=(x-2)^2(x+1)larrcolor(blue)"expand factors"`

`color(white)(f(x))=x^3-3x^2+4`

`rArrf'(x)=3x^2-6x`

`rArrf'(1)=3-6=-3<0`

`"since" 'f'(1)<0" then "f(x)" is decreasing at x = 1"`
graph{(x-2)^2(x+1) [-10, 10, -5, 5]}