a. find the open interval(s) on which the function is increasing and decreasing. b. identify the functions…

a. find the open interval(s) on which the function is increasing and decreasing. b. identify the functions local and absolute extreme values, if any, saying where they occur. $f(x)=x^{1 / 3}(x^{2}-4)$ a. on what open interval(s), if any, is the function increasing? select the correct choice below and if necessary, fill in the answer box(es) to complete your choice. a. the function is increasing on the open interval(s) (simplify your answer. type your answer in interval notation. type an exact answer, using radicals needed. use a comma to separate answers as needed.) b. the function is never increasing

a. find the open interval(s) on which the function is increasing and decreasing. b. identify the functions local and absolute extreme values, if any, saying where they occur. $f(x)=x^{1 / 3}(x^{2}-4)$ a. on what open interval(s), if any, is the function increasing? select the correct choice below and if necessary, fill in the answer box(es) to complete your choice. a. the function is increasing on the open interval(s) (simplify your answer. type your answer in interval notation. type an exact answer, using radicals needed. use a comma to separate answers as needed.) b. the function is never increasing

Answer

Explanation:

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

Use the product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = x^{1/3}) and (v=x^{2}-4). (u^\prime=\frac{1}{3}x^{-2/3}), (v^\prime = 2x) (f^\prime(x)=\frac{1}{3}x^{-2/3}(x^{2}-4)+x^{1/3}(2x)) Simplify: [ \begin{align*} f^\prime(x)&=\frac{x^{2}-4}{3x^{2/3}}+2x^{4/3}\ &=\frac{x^{2}-4 + 6x^{2}}{3x^{2/3}}\ &=\frac{7x^{2}-4}{3x^{2/3}} \end{align*} ]

Step2: Find critical points

Set (f^\prime(x)=0), then (\frac{7x^{2}-4}{3x^{2/3}} = 0). Since the denominator (3x^{2/3}\neq0) for (x\neq0), we solve (7x^{2}-4=0) (x^{2}=\frac{4}{7}), so (x=\pm\frac{2}{\sqrt{7}}=\pm\frac{2\sqrt{7}}{7}) The derivative (f^\prime(x)) is undefined at (x = 0), but (x = 0) is in the domain of (f(x)) (domain of (f(x)) is ((-\infty,\infty)))

Step3: Test intervals

  • For (x\in(-\infty,-\frac{2\sqrt{7}}{7})), let (x=-1), (f^\prime(-1)=\frac{7\times(- 1)^{2}-4}{3\times(-1)^{2/3}}=\frac{3}{3}=1>0)
  • For (x\in(-\frac{2\sqrt{7}}{7},0)), let (x =-\frac{1}{2}), (f^\prime(-\frac{1}{2})=\frac{7\times(\frac{1}{4})-4}{3\times(\frac{1}{2})^{2/3}}=\frac{\frac{7 - 16}{4}}{3\times(\frac{1}{2})^{2/3}}<0)
  • For (x\in(0,\frac{2\sqrt{7}}{7})), let (x=\frac{1}{2}), (f^\prime(\frac{1}{2})=\frac{7\times(\frac{1}{4})-4}{3\times(\frac{1}{2})^{2/3}}<0)
  • For (x\in(\frac{2\sqrt{7}}{7},\infty)), let (x = 1), (f^\prime(1)=\frac{7\times1^{2}-4}{3\times1^{2/3}} = 1>0)

Answer:

A. The function is increasing on the open interval(s) (\left(-\infty,-\frac{2\sqrt{7}}{7}\right)\cup\left(\frac{2\sqrt{7}}{7},\infty\right))