What happens if the second derivative is positive

Fundamentally, we are beginning to think about how a particular curve bends, with the natural comparison being made to lines, which don't bend at all. More than this, we want to understand how the bend in a function's graph is tied to behavior characterized by the first derivative of the function. On the leftmost curve in Figure 1. As we move from left to right, the slopes of those tangent lines will increase.

Therefore, the rate of change of the pictured function is increasing, and this explains why we say this function is increasing at an increasing rate. For the rightmost graph in Figure 1. This function is increasing at a decreasing rate. Similar options hold for how a function can decrease. Here we must be extra careful with our language, because decreasing functions involve negative slopes.

Negative numbers present an interesting tension between common language and mathematical language. 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. The derivative tells us if the original function is increasing or decreasing.

This second derivative also gives us information about our original function f. The second derivative gives us a mathematical way to tell how the graph of a function is curved.

The second derivative tells us if the original function is concave up or down. We can say that f is increasing or decreasing at an increasing rate. We can say that f is increasing or decreasing at a decreasing rate. At each of these times the velocity is positive and the particle is moving upward, increasing in height. An inflection point is a point on the graph of a function where the concavity of the function changes from concave up to down or from concave down to up.

The concavity changes at points b and g. This is especially important at points close to the critical stationary points. Critical points occur where the first derivative is 0. 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.

Similarly if the second derivative is negative, the graph is concave down. This is of particular interest at a critical point where the tangent line is flat and concavity tells us if we have a relative minimum or maximum.

For the designated function and point, determine if the graph has a local minimum, local maximum, or non-extreme point, or if the second derivative test fails. However if we look a the graph, we can see the curve is concave up everywhere, and that this point is a local minimum. However if we look a the graph, we can see this point is neither a local minimum or a local maximum.

It is a place where the graph switches from being concave up to being concave down. This is called an inflection point. We will use the second derivative test for finding maximums and minimums in the next chapter. The mathematical first and second derivatives are used in pricing various financial products and options that are also called derivatives. The first derivative is used to give a value to whether the underlying product has a price that goes up or down.

It looks at the slope of the pricing curve.