Understanding Critical Numbers and Monotonic Functions

The video tutorial explains how to find critical numbers and determine intervals where a function is increasing or decreasing. It uses the function f(x) = (5 - 8x)^5 as an example. The process involves finding where the first derivative is zero or undefined, using critical numbers to establish test intervals, and analyzing the sign of the derivative in each interval. The function is shown to be decreasing over its entire domain, making it a monotonic function.

What is the given function in the video?

• f(x) = (5 + 8x)^4

• f(x) = (5 - 8x)^4

• f(x) = (5 + 8x)^5

• f(x) = (5 - 8x)^5

What is the first step in finding intervals where a function is increasing or decreasing?

• Solve for x-intercepts

• Graph the function

• Find the critical numbers

• Find the second derivative

What is the derivative of the inner function 5 - 8x?

• 5

• 8

• -5

• -8

What is the critical number found for the function?

• x = 7/8

• x = 5/8

• x = 1/8

• x = 3/8

How is the number line divided using the critical number?

• Into two parts

• Into three parts

• Into four parts

• Into one part

What is the sign of the derivative in the interval where x < 5/8?

• Zero

• Negative

• Undefined

• Positive

What is the sign of the derivative in the interval where x > 5/8?

• Positive

• Negative

• Zero

• Undefined

What does it mean if a function is decreasing over its entire domain?

• The function is constant

• The function is monotonic

• The function is periodic

• The function is undefined

What is the term used to describe a function that is increasing or decreasing over its entire domain?

• Variable

• Constant

• Monotonic

• Periodic

What is the overall behavior of the function discussed in the video?

• Increasing

• Decreasing

• Constant

• Oscillating