multiple choice 2 points\nwhich of the following statements is true of ( f(x)=-x^{3}-6 x^{2}-9 x-2 ) ?\n( f…

multiple choice 2 points\nwhich of the following statements is true of ( f(x)=-x^{3}-6 x^{2}-9 x-2 ) ?\n( f ) is increasing on ( (-infty,-3) )\n( f ) is increasing on ( (-3,-1) )\n( f ) is decreasing for all real values\n( f ) is increasing on ( (-2, infty) )

multiple choice 2 points\nwhich of the following statements is true of ( f(x)=-x^{3}-6 x^{2}-9 x-2 ) ?\n( f ) is increasing on ( (-infty,-3) )\n( f ) is increasing on ( (-3,-1) )\n( f ) is decreasing for all real values\n( f ) is increasing on ( (-2, infty) )

Answer

Explanation:

Step1: Find the derivative of (f(x))

The function is (f(x)=-x^{3}-6x^{2}-9x - 2). Using the power rule ((x^{n})^\prime=nx^{n - 1}), the derivative (f^\prime(x)=-3x^{2}-12x - 9). Factor out (-3): (f^\prime(x)=-3(x^{2}+4x + 3)=-3(x + 1)(x+3)).

Step2: Determine the sign of (f^\prime(x))

Set (f^\prime(x)>0) (for increasing intervals) and (f^\prime(x)<0) (for decreasing intervals).

  • For (f^\prime(x)>0): (-3(x + 1)(x + 3)>0). Divide both sides by (-3) (inequality sign flips), we get ((x + 1)(x + 3)<0). The solution of ((x + 1)(x + 3)<0) is (x\in(-3,-1)).
  • For (f^\prime(x)<0): (-3(x + 1)(x + 3)<0). Divide both sides by (-3) (inequality sign flips), we get ((x + 1)(x + 3)>0). The solutions are (x\in(-\infty,-3)\cup(-1,\infty)).

Answer:

(f) is increasing on ((-3,-1))