Question

find the critical numbers of f determine the intervals on which the function is increasing or decreasing, and identify all relative extrema.
f(x)=(x-1)^2 (x+2)

Answer 1

@factor can you help me with this?

Answer 2

f'(x) = 2(x-1)(x+2)+(x-1)^2

Answer 3

so critical numbersa are x=1 and x=-2 right?

Answer 4

f'(x) = 2(x-1)(x+2)+(x-1)^2 = (x-1)(2x+4+x-1) = (x-1)(3x+3)
f'(x) = 3(x-1)(x+1)

The critical points are: x = 1 and x = -1

Answer 5

Analyze f'(x) in the intervals: (-infinity, -1), (-1, 1) and (1, infinity)
f'(x) = 3(x-1)(x+1)
f'(-2) = (-ve)(-ve) = +ve.
Therefore function is increasing in the interval (-infinity, -1).
f'(0) = (-ve)(+ve) = -ve
Therefore function is decreasing in the interval (-1, 1).
f'(2) = (+ve)(+ve) = +ve
Therefore function is increasing in the interval (1, infinity).
f(x) attains a maximum at x = -1 and a minimum at x = +1.

Answer 6

Thank you

Answer 7

You are welcome.