Question

For which interval(s) is the function increasing and decreasing?

y=3x^3-16x+2

a. increasing for −2.31 < x < 2.31; decreasing for x < −2.31 and x > 2.31

b. increasing for x lesser than -4/3 and ; decreasing for

c. increasing for and ; decreasing for

d. increasing for ; decreasing for and

Answer 1

Increasing:
`(-\infty,-(4)/(3))\cup((4)/(3),\infty)`

Decreasing:
`(-(4)/(3),(4)/(3))`

Explanation:

A continuous function increases where
`f'(x) > 0` and decreases where
`f'(x) < 0`, so let's find our critical points:

`f(x)=3x^3-16x+2\\f'(x)=9x^2-16\\`

`9x^2-16=0\\\\9x^2=16\\\\x^2=(16)/(9)\\\\x=-(4)/(3),(4)/(3)`

We'll use
`x=-2, x=0, \text{and } x=2` as test points to see how the derivative's behavior is around the critical points:

`f'(-2)=9(-2)^2-16=9(4)-16=36-16=20 > 0`

`f'(0)=9(0)^2-16=-16 < 0`

`f'(2)=9(2)^2-16=9(4)-16=36-16=20 > 0`

Hence, we can see that f(x) increases on the interval
`(-\infty,-(4)/(3))\cup((4)/(3),\infty)` and decreases on the interval
`(-(4)/(3),(4)/(3))`.