What if second derivative is negative

So, I'm confused over the interpretation of the second derivative. Why do you say this? Add a comment. Active Oldest Votes. Emilio Novati Emilio Novati Taussig N. Taussig Sign up or log in Sign up using Google. Sign up using Facebook.

Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post. Related 2. Hot Network Questions. Now consider the three graphs shown in Figure 1. Clearly the middle graph depicts a function decreasing at a constant rate. Now, on the first curve, draw a sequence of tangent lines. We see that the slopes of these lines get less and less negative as we move from left to right.

That means that the values of the first derivative, while all negative, are increasing, and thus we say that the leftmost curve is decreasing at an increasing rate. This leaves only the rightmost curve in Figure 1. For that function, the slopes of the tangent lines are negative throughout the pictured interval, but as we move from left to right, the slopes get more and more negative. Hence the slope of the curve is decreasing, and we say that the function is decreasing at a decreasing rate.

We now introduce the notion of concavity which provides simpler language to describe these behaviors. Concavity is linked to both the first and second derivatives of the function. In Figure 1. On the lefthand plot, where the function is concave up, observe that the tangent lines always lie below the curve itself, and the slopes of the tangent lines are increasing as we move from left to right.

Similarly, on the righthand plot in Figure 1. We state these most recent observations formally as the definitions of the terms concave up and concave down. Acceleration is defined to be the instantaneous rate of change of velocity, as the acceleration of an object measures the rate at which the velocity of the object is changing.

Using only the words increasing , decreasing , constant , concave up , concave down , and linear , complete the following sentences. Exploring the context of position, velocity, and acceleration is an excellent way to understand how a function, its first derivative, and its second derivative are related to one another. In Activity 1. Those values and more are provided in the second table below, along with several others computed in the same way. To determine the type of stationary point, calculate the second derivative at each value of x.

To determine the y -coordinate of the point, calculate the value of the function for each value of x. Hence 0,0 is a local maximum, -3, is a local minimum and 5, is a local minimum. If x then f " x is positive and the graph is concave upward. When - x f " x is negative and the graph is concave downward. To determine the y -coordinate of these points of inflection, calculate the value of the function for each value of x.