Differentiate f(x) = x³-6x²+9x+1
[tex] \frac{d}{dx} ( x^{3} -6x ^{2} +9x+1)[/tex]
Solution for each term:
[tex] \frac{d}{dx} (x^{3} ) = (3)x^{3-1} = 3 x^{2} [/tex]
[tex] \frac{d}{dx}(-6(2)x ^{2-1} ) = -12x[/tex]
[tex] \frac{d}{dx} (9(1)x^{1-1} ) = 9[/tex]
[tex] \frac{d}{dx} (1) = 0[/tex]
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.)