Given f(x)=x^3+6x^2+12x+9
where does f(x) Increase and when does it decrease?
where does f(x) Increase and when does it decrease?
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f(x) increases where f '(x) > 0 and decreases where f '(x) < 0.
f '(x) = 3x^2 + 12x + 12
3x^2 + 12x + 12 > 0
x^2 + 4x + 4 > 0
(x + 2)^2 > 0 which is true for all x
3x^2 + 12x + 12 < 0
x^2 + 4x + 4 < 0
(x + 2)^2 < 0 which has no solutions
Therefore, f(x) is increasing on (-∞, ∞).
f '(x) = 3x^2 + 12x + 12
3x^2 + 12x + 12 > 0
x^2 + 4x + 4 > 0
(x + 2)^2 > 0 which is true for all x
3x^2 + 12x + 12 < 0
x^2 + 4x + 4 < 0
(x + 2)^2 < 0 which has no solutions
Therefore, f(x) is increasing on (-∞, ∞).