Consider the following. (If an answer does not exist, enter DNE.) f '(x) = 9x^2 − 4

Question

(a)

Find the open intervals on which f ′(x) is increasing or decreasing. (Enter your answers using interval notation.)

increasing:

decreasing:

(b)

Find the open intervals on which the graph of f is concave upward or concave downward. (Enter your answers using interval notation.)

concave upward:

concave downward:

(c)

Find the x-values of the relative extrema of f. (Enter your answers as a comma-separated list.)

relative maximum x =

relative minimum x =

Find the x-values of the inflection points of f. (Enter your answers as a comma-separated list.)

x =

Answer 1

(a)

Find the open intervals on which f ′(x) is increasing or decreasing. (Enter your answers using interval notation.)

Take the first derivative. Set it equal to zero and solve for the critical value.

y'=18x

0=18x

x=0 is the critical value

Plug in test values

x=-1

(-1)(18) = negative so decreasing

x=1

(1)(18) = positive so increasing

increasing: (0,∞)

decreasing: (-∞,0)

(b)

Find the open intervals on which the graph of f is concave upward or concave downward. (Enter your answers using interval notation.)

y" = 18 always positive so always concave up.

concave upward: (-∞,∞)

concave downward: none

(c)

Find the x-values of the relative extrema of f. (Enter your answers as a comma-separated list.)

The first derivative goes from negative to positive, so x=0 is a relative min.

relative maximum x = none

relative minimum x = 0

Find the x-values of the inflection points of f. (Enter your answers as a comma-separated list.)

x = none

This is always concave up, so there are no points where it changes concavity.