which of the following statements is true of $f(x)=-x^{3}-6x^{2}-9x - 2$?\n$f$ is increasing on…

which of the following statements is true of $f(x)=-x^{3}-6x^{2}-9x - 2$?\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 $(-2,\\infty)$

which of the following statements is true of $f(x)=-x^{3}-6x^{2}-9x - 2$?\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 the function

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 the derivative

Set (f^\prime(x)=0), then (-3(x + 1)(x + 3)=0), so (x=-3) or (x=-1). We consider the intervals ((-\infty,-3)), ((-3,-1)) and ((-1,\infty)). Take a test - point in ((-\infty,-3)), say (x=-4). Then (f^\prime(-4)=-3(-4 + 1)(-4 + 3)=-3\times(-3)\times(-1)=-9<0). Take a test - point in ((-3,-1)), say (x=-2). Then (f^\prime(-2)=-3(-2 + 1)(-2 + 3)=-3\times(-1)\times1 = 3>0). Take a test - point in ((-1,\infty)), say (x=0). Then (f^\prime(0)=-3(0 + 1)(0 + 3)=-9<0).

Answer:

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