`f(x) = (x+2)^2(x- 1)` Find the critical numbers, open intervals on which the function is increasing or decreasing, apply first derivative test to identify all relative extrema.

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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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This function is defined everywhere, its critical numbers are where f'(x)=0.

f'(x)=2*(x+2)*(x-1) + (x+2)^2  =

(x+2)*(2x-2+x+2) = (x+2)*(3x).

This is =0 at x=-2 and x=0. Also f'(x) is negative for x in (-2, 0) and positive in `(-oo, -2) uuu (0, +oo).` The function f is decreases and increases correspondently. Therefore x=-2 is the local maximum and x=0 is a local minimum.