(a) determine the critical numbers. select the correct choice below and, if necessary, fill in the answer…

(a) determine the critical numbers. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical number(s) is/are (type an integer or a simplified fraction. use a comma to separate answers as needed.) b. there are no critical numbers. (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. a. increasing on (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.) b. never increasing (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. a. decreasing on (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.) b. never decreasing

(a) determine the critical numbers. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical number(s) is/are (type an integer or a simplified fraction. use a comma to separate answers as needed.) b. there are no critical numbers. (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. a. increasing on (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.) b. never increasing (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. a. decreasing on (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.) b. never decreasing

Answer

Explanation:

Step1: Find the derivative of the function

Given ( f(x)=\frac{2}{3}x^{3}-x^{2}-14x - 12). Using the power rule ((x^n)^\prime=nx^{n - 1}), we have (f^\prime(x)=2x^{2}-2x - 14). Set (f^\prime(x) = 0), so (2x^{2}-2x - 14=0). Divide through by 2: (x^{2}-x - 7=0). Using the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}) for (ax^{2}+bx + c = 0)((a = 1,b=-1,c=-7)), we get (x=\frac{1\pm\sqrt{1+28}}{2}=\frac{1\pm\sqrt{29}}{2}).

Step2: Determine the intervals of increase and decrease

We have critical points (x_1=\frac{1-\sqrt{29}}{2}\approx - 2.19\) and (x_2=\frac{1+\sqrt{29}}{2\approx3.19). Choose test points:

  • For (x=-\ 3) (in the interval ((-\infty,\frac{1 -\sqrt{29}}{2}))), (f^\prime(-3)=2\times(-3)^{2}-2\times(-3)-14=18 + 6-14 = 10>0).
  • For (x = 0) (in the interval ((\frac{1-\sqrt{29}}{2},\frac{1+\sqrt{29}}{2}))), (f^\prime(0)=2\times0^{2}-2\times0-14=-14<0).
  • For (x = 4) (in the interval \((\frac{1+\sqrt{29}}{2\infty})\), (f^\prime(4)=2\times4^{2}-2\times4-14=32-8 - 14 = 10>0).

Answer:

(a) The critical numbers are (x=\frac{1-\sqrt{29}}{2}) and (x=\frac{1+\sqrt{29}}{2}). (b) The function is increasing on ((-\infty,\frac{1-\sqrt{29}}{2})\cup(\frac{1+\sqrt{29}}{2},\infty)). (c) The function is decreasing on ((\frac{1-\sqrt{29}}{2},\frac{1+\sqrt{29}}{2})).