Analyzing Function Graphs with the Second Derivative

Concavity and the second derivative test So far, we have seen that the first derivative implies an increasing or decreasing nature for a function. Now, we'll look at the second derivative and see what it tells us about a given function. "What does the sign of the second derivative tell us about our function?" The sign of the second derivative tells us if the slope of our function is increasing or decreasing. If it's positive, then the slope is increasing. If it's negative, then the slope is decreasing. The second derivative of a function f(x) is denoted as f'(x) and its formula can be derived from that of f(x). Let us go through and take a look at the derivatives of x squared minus 3. This appears as 2x; the derivative of that would be 2, it looks like this; its derivative would be 2, as well. If we move through this progression, we see that the derivative of 3 minus x squared is negative 2x, which has a negative slope like so. The derivative of negative 2x would be negative 2; it looks like this. So what can we say about the sign of g in this case? In order for g to have a positive second derivative, f must have a positive derivative. And in order for g to have a negative second derivative, f must have a negative derivative. What does this tell us about our function? What is the main difference between these two functions? Well, this one opens up.And then this guy over here opens down.So it turns out that the direction of open is what is determined by the sign of the second derivative. Well, if f is always increasing, then it always has positive slope, which means that its derivative, f double prime, is always positive, which then ties it back to f and that it opens up. And the official term for "opens up" is the concavity.So this is concave up, and this would then be concave down. So let's put that together. f is concave up when f double prime is positive, and then f is concave down when f double prime is negative. Now that we have a second derivative, we can get more information about the graph of f. To do this, we will simply sketch a random graph of f to see if there are any relative extrema. Thus, if a first derivative changes from positive to negative, then we have a relative maximum, and if the first derivative changes from negative to positive, we have a relative minimum. So let's take a look at this relative max. To find the relative maximum of a function, we look at the second derivative and see which value of x has the greatest rate of increase. For this function, we know that f double prime is less than 0, because all of this region is concave down. In this case, we have a horizontal tangent. The derivative is equal to zero, so we know that the point is at a relative maximum. Similarly, if the derivative is equal to negative one, we know that it's at a relative minimum. It depends on this portion of the graph, which is concave up, so f'(x) must be greater than 0. We know that this point is a critical point because there is a horizontal tangent there. So if you have a critical point on a portion of the graph that is concave up, then you have a relative minimum. You will notice that here is a circled element; it's concave up, concave down. In this area, we have a change in concavity. This is known as a point of inflection. Well, you can determine the presence of a point of inflection by finding a sign change in the second derivative. You have seen that we have used derivatives, first and second, to describe numerous features about a graph. Now this section has been very vocabulary-intensive.