Sketch a graph of a function f with the following properties.

Question

f' < 0 and f" < 0, for x < 3

Accepted Answer

Hello. In this video, we are going to be answering the following graph problem. We are told that a function F of X has the following properties. Its derivative is greater than 0, and its double derivative is greater than 0 for the value X greater than 4. Which of the following is a possible graph of F of X? Now, let's just go ahead and list down what the problem tells us. We have some generic F of X. We are told that F of X is greater than 0. Now, what does it mean for a graph's derivative to be greater than 0? Well, when a graph's derivative is greater than 0, what that means is that F of X is increasing. When Its derivative F of X is greater than 0. Since we are focused on the values of X that are greater than 4, and we know that the graph F of X is increasing for its derivative greater than 0, that means that F of X will be increasing. At the value X greater than 4. Now, we need to pick a possible graph that matches this first description. So, let's go ahead and label the point 4.0 on each of the given graphs. That point will be located here. Now, which of these graphs will have the graph F of X increasing from the value 4 to infinity? While the graphs for A and B don't match this description, and the reason why is because the graph is decreasing for both of these graphs after the value of 4. So A and B will not be a potential answer, but C and D has the graph of F of X increasing. After the value of 4. So these two will be possible graphs to F of X. Now let's go ahead and review the 2nd condition given to us. We are told that the double derivative F of X is greater than 0. When the double derivative is greater than 0, that tells us that F of X is concave up. On the values from X is greater than 4 to infinity. So, which of these conditions from C and D will match concave up? Well, the easiest way to remember concave up is to imagine a glass of water. If the characteristics of a graph look like an upright cup of water, then that's how you know the conditions of the graph will be concave up. And that condition of the graph is seen in the in option D. On the value from 4 to infinity, the graph is increasing slowly at an exponential rate. This matches the condition of an increasing glass of water. Now, why would C not be concave up? Well, the reason why C is concave up is because it's a continuous constant increase, whereas the increase in D increases over time instead of it being a constant increase. What this means is that D is going to be the solution to this problem, since it matches the conditions of F of X increasing after 4, and being concave up after 4. So, I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Sketch a graph of a function f with the following properties.

f' < 0 and f" < 0, for x < 3

Key Concepts

First Derivative Test

Second Derivative Test

Graph Behavior

Watch next