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.

Explanation:

Step1: Find the derivative of the function

The function is \( y = -3x + 16 \). The derivative \( y' \) of a linear function \( y = mx + b \) is the slope \( m \). So, \( y'=-3 \).

Step2: Analyze critical numbers

Critical numbers occur where the derivative is zero or undefined. The derivative \( y' = - 3 \) is a constant (defined for all real numbers) and never zero. So, there are no critical numbers.

Step3: Analyze increasing intervals

A function is increasing on an interval if its derivative is positive on that interval. Since \( y'=-3<0 \) for all real numbers \( x \), the derivative is never positive. So, there are no intervals where the function is increasing.

Answer:

(a) B. There are no critical numbers.
(b) B. There are no intervals where the function is increasing.