8.2.3 Regions Where a Function Increases or Decreases

Question 1

Regions Where a Function Increases or Decreases

Answer 1

• Critical points divide the curve of a function into regions where the function either increases or decreases.
• On an interval, the sign of the first derivative of a function indicates whether that function is increasing or decreasing

Question 2

note

Answer 2

• Consider the regions of this graph that are defined by the critical points.
• Between any two adjacent critical points the curve behaves in only one way. It is either increasing or it is decreasing.
• Notice that there is no way for the behavior of the curve to change without it either leveling off or making a sharp turn. When either situation occurs, there is a critical point, indicating a possible change from increasing to decreasing, or decreasing to increasing.
• However, a critical point does not always force a curve to change its behavior. For example, x 5 is a critical point, but the function is increasing on both the right and the left.
• Recall that a critical point of a function is a point where the derivative of the function is either zero or undefined. Thus it is possible to have a point where a function is continuous but not differentiable.
• Here is an example of a function whose critical point you have already determined.
• To study the behavior of the curve, make a sign chart. To keep it simple, just label the critical point.
• Next, choose any x-value that is less than the critical value. Here an easy value is x = –1. Since the derivative at x = –1 is negative, all the values to the left of the critical point will make the derivative negative. The function is decreasing on that region. Mark a down-arrow on the sign chart.
• You must also choose an x-value to the right of the critical point. Here an easy value is x = 0. Since that value makes the derivative positive, the function is increasing on that region. Mark an up-arrow on the sign chart.
• Since the function is decreasing to the left of the critical point and increasing to the right, you can conclude that the curve reaches a minimum point at the critical point.

Question 3

Find the interval(s) where the function y=(x^2−9)^2/3 is decreasing.

Answer 3

x < −3 or 0 < x < 3

Question 4

Based on this graph of f ‘(x), where is the function f(x) increasing?

Answer 4

−1 < x < 1

Question 5

Find the interval(s) where the function g(x)=x^3−3x^2+2 is decreasing.

Answer 5

0 < x < 2

Question 6

Suppose you know that a continuous function f (x) is increasing at x = 2, and decreasing at x = 4, and that x = 3 is a critical point. Which of the following could describe f ′(3)?

Answer 6

f ′(3) does not exist.

Question 7

Find the interval(s) where the function f(x)=1/x^2 is increasing.

Answer 7

x < 0

Question 8

Suppose you are told that H ′(x) is positive on the intervals −3 < x < −1 and x > 5 (and nowhere else). Which of the following could be a graph of H (x)?

Answer 8

Since H ′(x) is positive on the intervals −3 < x < − 1 and x > 5, H (x) will be increasing on those intervals. Graph C describes such a function

Question 9

Suppose f(x)is a continuous differentiable function and you are given that f′′(x) is always positive. Which of the following statements must be true?

Answer 9

f′(x) is always increasing

Question 10

On what interval is this function increasing?

Answer 10

x < 0 or x > 0

Question 11

On what interval is the function f(x)=xlnx+2 decreasing?

Answer 11

0

Question 12

Find the interval(s) where f (x) = x ^3 − 3x is increasing.

Answer 12

x < −1 or x > 1