Consider the following function.

Question

Consider the following function.

f(x) = −4x³ + 12x + 3

(a)

Find the critical numbers of f. (Enter your answers as a comma-separated list.)

x =

(b)

Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)

increasing

decreasing

Answer 1

We begin by taking the first derivative of this function, which gives us f '(x) = -12x² + 12. Factoring out a -12 and setting this = to 0, we have -12( x² - 1) = 0 → -12 ( x + 1 )(x - 1 ), so our critical points are -1, 1.

To determine where the function is increasing or decreasing, we can partition a number line into 3 sections with the dividers at x = -1 and x = 1, choosing random values of x in each partition to see where f'(x) is positive and where f'(x) is negative. For values, let's use, say, x = -2, x = 0, and x = 2. Inserting each value into the formula for f '(x), here's what we see:

f '(-2) = -36 ( a negative value so f(x) is decreasing here );

f '(0) = 12 ( a positive value so f(x) is increasing here) ;

f '(2) = -36 ( so again, f(x) is decreasing here).

So using interval notation, here's what we have: decreasing on ( -∞, -1 ) and ( 1, ∞ );

increasing on ( -1, 1 )