Understanding Increasing and Decreasing Functions

The video tutorial explains how to determine the intervals where a function is increasing or decreasing. It covers the use of the first derivative to find critical points and create sign charts. The tutorial includes examples with quadratic, cubic, and absolute value functions, demonstrating how to analyze each type to find intervals of increase and decrease. The video emphasizes understanding the graphical representation and algebraic manipulation to identify these intervals.

What does a positive first derivative indicate about a function's behavior?

• The function is decreasing.

• The function is increasing.

• The function has a maximum point.

• The function is constant.

In the quadratic function example, what is the critical number found?

• 3/2

• 2

• 0

• 1

For the quadratic function f(x) = x^2 - 3x + 1, in which interval is the function increasing?

• (-∞, 3/2)

• (0, 3/2)

• (-∞, 0)

• (3/2, ∞)

What is the first step in analyzing the cubic function f(x) = x^3 - 9x^2 + 24x?

• Set the function equal to zero.

• Graph the function.

• Find the second derivative.

• Find the first derivative.

In the cubic function example, what are the critical numbers found?

• 0 and 5

• 2 and 4

• 1 and 3

• 3 and 6

For the cubic function, in which interval is the function decreasing?

• (4, ∞)

• (-∞, 2)

• (0, 3)

• (2, 4)

How is the graph of the absolute value function shifted when considering |x - 3|?

• 3 units to the left

• 3 units to the right

• 3 units up

• 3 units down

For the absolute value function, where is the function increasing?

• (3, 0)

• (0, 3)

• (-∞, 3)

• (3, ∞)

In the linear function example, what is the vertex of the function 3x + 4?

• 0

• -4/3

• 3

• 4/3

For the linear function, in which interval is the function decreasing?

• (3, ∞)

• (-4/3, ∞)

• (0, ∞)

• (-∞, -4/3)