the second derivative. Whether making such a change to the notation is sufficiently helpful to be worth the trouble is still under debate. Example Given the function the derivative of is the function The second derivative of is the derivative of , namely Relation to the graph Concavity The second derivative of a function can be used to determine the concavity of the graph of . A function whose second derivative is positive will be concave up (also referred to as convex), meaning that the tangent line will lie below the graph of the function. Similarly, a function whose second derivative is negative will be concave down (also simply called concave), and its tangent lines will lie above the graph of the function. Inflection points If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection. Second derivative test The relation between the second derivative and the graph can be used to test whether a stationary point for a function (i.e., a point where ) is a local maximum or a local minimum. Specifically, If , then has a local maximum at . If , then has a local minimum at . If , the second derivative test says nothing about the point , a possible inflection point. The reason the second derivative produces these results can be seen by way of a real-world analogy. Consider a vehicle that at first is moving forward at a great velocity, but with a negative acceleration. Clearly, the position of the vehicle at the point where the velocity reaches zero will be the maximum distance from the starting position – after this time, the velocity will become negative and the vehicle will reverse. The same is true for the minimum, with a vehicle that at first has a very negative velocity but positive acceleration. Limit It is possible to write a single limit for the second derivative: The limit is called the second symmetric derivative. Note that the second symmetric derivative may exist even when the (usual) second derivative does not. The expression on the right