4 multiple choice 4 points which of the following statements is true of ( f(x)=-x^{3}-6 x^{2}-9 x - 2 )? ( f…

4 multiple choice 4 points which of the following statements is true of ( f(x)=-x^{3}-6 x^{2}-9 x - 2 )? ( f ) is increasing on ( (-2, infty) ) ( f ) is decreasing on ( (-3,-1) ) ( f ) is decreasing for all real values ( f ) is increasing on ( (-3,-1) ) 5 multiple choice 4 points let ( f ) be defined by ( f(x)=left(x^{2}-1\right)^{3} ) for all real numbers ( x ). for what values of ( x ) is the function increasing? ( (-1,1) ) ( 0, infty) ) ( (-1,0 ) ( (1, infty) )

4 multiple choice 4 points which of the following statements is true of ( f(x)=-x^{3}-6 x^{2}-9 x - 2 )? ( f ) is increasing on ( (-2, infty) ) ( f ) is decreasing on ( (-3,-1) ) ( f ) is decreasing for all real values ( f ) is increasing on ( (-3,-1) ) 5 multiple choice 4 points let ( f ) be defined by ( f(x)=left(x^{2}-1\right)^{3} ) for all real numbers ( x ). for what values of ( x ) is the function increasing? ( (-1,1) ) ( 0, infty) ) ( (-1,0 ) ( (1, infty) )

Answer

Question 4

Explanation:

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

Using the power rule ((x^n)^\prime=nx^{n - 1}), if (f(x)=-x^{3}-6x^{2}-9x - 2), then (f^\prime(x)=-3x^{2}-12x - 9=-3(x^{2}+4x + 3)=-3(x + 1)(x+3))

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

Set (f^\prime(x)=0), we get (x=-3) and (x=-1). We consider the intervals ((-\infty,-3)), ((-3,-1)) and ((-1,\infty))

  • For (x\in(-\infty,-3)), let (x=-4), then (f^\prime(-4)=-3(-4 + 1)(-4+3)=-3\times(-3)\times(-1)=-9<0)
  • For (x\in(-3,-1)), let (x=-2), then (f^\prime(-2)=-3(-2 + 1)(-2 + 3)=-3\times(-1)\times1 = 3>0)
  • For (x\in(-1,\infty)), let (x=0), then (f^\prime(0)=-3(0 + 1)(0+3)=-9<0)

Answer:

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

Question 5

Explanation:

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

Using the chain - rule ((u^n)^\prime=nu^{n - 1}\cdot u^\prime), if (u = x^{2}-1) and (n = 3), then (f^\prime(x)=3(x^{2}-1)^{2}\cdot2x=6x(x^{2}-1)^{2}=6x(x - 1)^{2}(x + 1)^{2})

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

Since ((x - 1)^{2}\geq0) and ((x + 1)^{2}\geq0) for all real (x). The sign of (f^\prime(x)) is determined by the factor (6x).

  • When (x>0), (f^\prime(x)\geq0) (equality holds when (x = 1) or (x=-1))

Answer:

([0,\infty))