Question

for the following function, find (a) the critical numbers, (b) the open intervals where the function is increasing, and (c) the open intervals where it is decreasing.

$y = -3x + 16$

(a) identify the critical numbers of the function. 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) identify the open intervals where the function is increasing. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the function is increasing on the interval(s) .
(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. there are no intervals where the function is increasing.

(c) identify the open intervals where the function is decreasing. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the function is decreasing on the interval(s) $(-infty, infty)$.
(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. there are no intervals where the function is decreasing.

Explanation:

Step1: Recall Critical Number Definition

A critical number of a function \( y = f(x) \) is a point \( c \) where \( f'(c) = 0 \) or \( f'(c) \) is undefined. For \( y=-3x + 16 \), the derivative \( y'=-3 \), which is a constant (defined everywhere and never zero). So, no critical numbers.

Step2: Analyze Increasing Intervals

A function is increasing on an interval if \( y'>0 \) there. Since \( y'=-3<0 \) for all \( x \), there are no intervals where \( y' > 0 \), so no increasing intervals.

Step3: Analyze Decreasing Intervals

A function is decreasing on an interval if \( y'<0 \) there. Since \( y'=-3<0 \) for all real numbers \( x \), the interval is \( (-\infty, \infty) \) (note: the original answer had a typo, it should be \( (-\infty, \infty) \), not \( (\infty, -\infty) \)).

Answer:

(a) B. There are no critical numbers.
(b) B. There are no intervals where the function is increasing.
(c) A. The function is decreasing on the interval(s) \((-\infty, \infty)\).