for the function below, find (a) the critical numbers; (b) the open intervals where the function is…

for the function below, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.\n$f(x)=\\frac{4}{3}x^{3}-4x^{2}-252x - 20$\n(a) determine the critical numbers. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the critical number(s) is/are \n(type an integer or a simplified fraction. use a comma to separate answers as needed.)\nb. there are no critical numbers.\n(b) on which intervals is the function increasing? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. increasing on \n(type your answer in interval notation. simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. never increasing\n(c) on which intervals is the function decreasing? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. decreasing on \n(type your answer in interval notation. simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. never decreasing

for the function below, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.\n$f(x)=\\frac{4}{3}x^{3}-4x^{2}-252x - 20$\n(a) determine the critical numbers. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the critical number(s) is/are \n(type an integer or a simplified fraction. use a comma to separate answers as needed.)\nb. there are no critical numbers.\n(b) on which intervals is the function increasing? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. increasing on \n(type your answer in interval notation. simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. never increasing\n(c) on which intervals is the function decreasing? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. decreasing on \n(type your answer in interval notation. simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. never decreasing

Answer

Explanation:

Step1: Find the derivative of the function

Given ( f(x)=\frac{4}{3}x^{3}-4x^{2}-252x - 20). Using the power rule ( (x^{n})^\prime=nx^{n - 1}), we have ( f^\prime(x)=4x^{2}-8x - 252). Factor out the common factor: ( f^\prime(x)=4(x^{2}-2x - 63)). Factor the quadratic expression: ( f^\prime(x)=4(x - 9)(x+7)).

Step2: Find the critical numbers

Set ( f^\prime(x) = 0), then (4(x - 9)(x + 7)=0). Using the zero - product property (ab = 0\Rightarrow a=0) or (b = 0), we get (x-9=0) or (x + 7=0). So the critical numbers are (x=-7) and (x = 9).

Step3: Determine the intervals of increase and decrease

We use test points in the intervals ((-\infty,-7)), ((-7,9)) and ((9,\infty)).

  • For the interval ((-\infty,-7)), let (x=-8). Then (f^\prime(-8)=4(-8 - 9)(-8 + 7)=4\times(-17)\times(-1)=68>0). So (f(x)) is increasing on ((-\infty,-7)).
  • For the interval ((-7,9)), let (x = 0). Then (f^\prime(0)=4(0 - 9)(0 + 7)=4\times(-9)\times7=-252<0). So (f(x)) is decreasing on ((-7,9)).
  • For the interval ((9,\infty)), let (x = 10). Then (f^\prime(10)=4(10 - 9)(10 + 7)=4\times1\times17 = 68>0). So (f(x)) is increasing on ((9,\infty)).

Answer:

(a) The critical numbers are (x=-7) and (x = 9). (b) The function is increasing on ((-\infty,-7)\cup(9,\infty)). (c) The function is decreasing on ((-7,9)).