find the intervals on which ( f(x) ) is increasing, the intervals on which ( f(x) ) is decreasing, and the…

find the intervals on which ( f(x) ) is increasing, the intervals on which ( f(x) ) is decreasing, and the local extrema.\n\n( f(x)=2 x^{4}+16 x^{3}+33 )\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the function is increasing on\n(type your answer in interval notation. type integers or simplified fractions. use a comma to separate answers as needed.)\nb. the function is never increasing.

find the intervals on which ( f(x) ) is increasing, the intervals on which ( f(x) ) is decreasing, and the local extrema.\n\n( f(x)=2 x^{4}+16 x^{3}+33 )\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the function is increasing on\n(type your answer in interval notation. type integers or simplified fractions. use a comma to separate answers as needed.)\nb. the function is never increasing.

Answer

Explanation:

Step1: Find the derivative of the function

The derivative of (f(x)=2x^{4}+16x^{3}+33) is (f^\prime(x)=8x^{3}+48x^{2}) (using the power rule ((x^n)^\prime = nx^{n - 1})). Factor out (8x^{2}): (f^\prime(x)=8x^{2}(x + 6)).

Step2: Find the critical points

Set (f^\prime(x)=0). Since (8x^{2}(x + 6)=0), then (x = 0) or (x=-6) (because (8x^{2}=0) gives (x = 0) and (x + 6=0) gives (x=-6)).

Step3: Test the intervals

  • For (x<-6), let (x=-7). Then (f^\prime(-7)=8\times(-7)^{2}\times(-7 + 6)=8\times49\times(-1)<0).
  • For (-6<x<0), let (x=-1). Then (f^\prime(-1)=8\times(-1)^{2}\times(-1 + 6)=8\times1\times5>0).
  • For (x>0), let (x = 1). Then (f^\prime(1)=8\times1^{2}\times(1 + 6)=8\times1\times7>0).

The function (f(x)) is increasing when (f^\prime(x)>0). The intervals where (f^\prime(x)>0) are ((-6,0)\cup(0,\infty)) (the function (y = f^\prime(x)) is positive for (x>-6) except (x = 0), but at (x = 0) the function (f(x)) has a horizontal tangent and the function is still increasing on either side of (x = 0) in the sense of the overall trend).

Answer:

A. The function is increasing on ((-6,\infty))