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.

Explanation:

Step1: Recall critical number definition

A critical number of a function \( y = f(x) \) is a number \( c \) in the domain of \( f \) where \( f'(c) = 0 \) or \( f'(c) \) does not exist.

Step2: Find the derivative of \( y = -3x + 16 \)

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

Step3: Analyze the derivative

The derivative \( y' = -3 \) is a constant (it exists for all real numbers) and is never equal to 0. So, there are no values of \( x \) for which \( y' = 0 \) or \( y' \) does not exist.

Answer:

B. There are no critical numbers.