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2015-10-18T22:45:34+08:00

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Differentiate f(x) = x³-6x²+9x+1
 \frac{d}{dx} ( x^{3} -6x ^{2} +9x+1)

Solution for each term:
 \frac{d}{dx} (x^{3} ) = (3)x^{3-1} = 3 x^{2}

 \frac{d}{dx}(-6(2)x ^{2-1} ) = -12x

 \frac{d}{dx} (9(1)x^{1-1} ) = 9

  \frac{d}{dx} (1) = 0

Simplify:
f(x)=(3x²-12x+9) ⇒ 3 (x²-4x+3) ⇒ 3(x-3)(x-1)

Stationary Points:
x-3 = 0               x-1 = 0
x = 3                  x = 1

INTERVALS:
(-∞,1)   (1,3)   (3,∞)

Increasing at intervals (-∞,1) and (3,∞)

Decreasing at interval (1,3)

(Note:  It's easier to solve for the intervals with derivatives than by factoring or zero theorem for the given function, avoiding the irrational complex numbers not necessary to what you required.)

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