8.3.1 Concavity and Inflection Points

Question 1

Concavity and Inflection Points

Answer 1

• The concavity of a graph can be determined by using the second derivative.
• If the second derivative of a function is positive on a given interval, then the graph of the function is concave up on that interval. If the second derivative of a function is negative on a given interval, then the graph of the function is concave down on that interval.
• Points where the graph changes concavity are called inflection points.

Question 2

note

Answer 2

• Given a function, you can determine where it is increasing and where it is decreasing. The next property to examine is curvature, or concavity.
• Notice that the graph on the left is decreasing and then increasing, but it is curved upward. The graph is said to be concave up. It resembles the outline of a coffee cup that is upright.
• On the right, the graph is curved downward. It is said to be concave down. This time it resembles the outline of an overturned coffee cup.
• To determine the concavity of a function, you will need to study the behavior of its derivative.
• Notice that the slopes of the tangent lines start out negative, then become zero, and finally become positive. They are increasing. Therefore the derivative is increasing.
• Another way of saying that the derivative is increasing is to say that the second derivative is positive.
• You can conclude that if the second derivative is positive, the function is concave up. Similarly, if the second derivative is negative, the function is concave down.
• This graph is concave down on the left and concave up on the right. The point where the concavity changes is called an inflection point.
• An inflection point can only occur where the second
  derivative is zero or undefined.

Question 3

Suppose you are told that in the interval a < x < b, the slope of the function h (x) is decreasing as x increases. Is h (x) concave up or concave down in this interval?

Answer 3

Concave down

Question 4

Given the graph of g(x), find the intervals where g(x) is concave up.

Answer 4

x < x1

Question 5

Is the function h (x) = 2 cos x + sin^2 x concave up or down at the point x=π?

Answer 5

Concave up

Question 6

Suppose you are given that f ″(x) < 0 on the intervals x < −1 and x > 1 (and nowhere else). Which of the following could be a graph of f (x)?

Answer 6

Graph C describes a function which is concave down on the appropriate intervals. The following is a good rule of thumb: If the graph is shaped like a bowl on an interval, then the function is concave up there. If the graph is shaped like an upside-down bowl, then the function is concave down there.

To be more precise, notice that on the intervals x < −1 and x > 1, the slope of the tangent line to the graph decreases as x increases. That means that the function is concave down on the intervals x < −1 and x > 1.

Question 7

Suppose you are given the function s (t) = t ^3 − 5t − 1. Is s (t) concave up or concave down at t = 2?

Answer 7

Concave up

Question 8

If the graph of the second derivative of f(x)is shown, on which of the following intervals is f(x) concave up?

Answer 8

(q,s)

Question 9

If the graph of the second derivative is shown, on which of the following intervals is f(x) concave down?

Answer 9

(s,t)