Question

How do you find the intervals of increasing and decreasing using the first derivative given `y=(x+2)^2(x-1)`?

Answer 1

Using the product rule:

`dy/dx = 2(x+2)(x-1) +(x+2)^2 = (x+2)(2x-2+x+2) = 3x(x+2)`

So:

`dy/dx < 0` for `x in (-2,0)`

`dy/dx > 0` for `in in (-oo,-2) uu (0,+oo)`

which means that `y(x)` is increasing from `-oo` to `x=-2` where it has a local maximum, decreasing from `x=-2` to `x=0` where it has a local minimum, and again increasing from `x=0` to `+oo`.

graph{(x+2)^2(x-1) [-4, 3, -10, 10]}