# Why 2nd derivative can find concavity

• vanmaiden
In summary, the process of finding critical points involves setting the first derivative to zero or undefined, and then using the second derivative to determine the concavity of the graph. The second derivative is positive for a concave upward graph and negative for a concave downward graph. The second derivative is also the rate of change of the first derivative, which explains why it can determine the concavity of the graph. Differences in the values of the second derivative at different points indicate changes in concavity.
vanmaiden

## Homework Statement

I know one can find the critical points by taking whatever points make the first derivative zero or undefined. I also know that when you put the critical points into the second derivative the positivity or negativity of the number will tell you whether the graph will be concave up or down. What gets me is that the critical point you put into the 2nd derivative has a slope of zero. How can you find out the concavity from a point with a slope of zero?

## The Attempt at a Solution

I have been thinking about this for awhile and it stumps me on why this way to find the concavity actually works.

Last edited:
You can't! That's not true. For example, the function $f(x)= x^2$ is "concave upward" for all x but the second derivative is 2, not 0, for all x. A function is "concave upward" as long as the second derivative is positive, concave downward if the second derivative is negative. The second derivative is 0 where the concavity changes.

HallsofIvy said:
A function is "concave upward" as long as the second derivative is positive, concave downward if the second derivative is negative.

i see, but what makes the second derivative have this property?

The second derivative is, of course, the rate of change of the first derivative. If the second derivative is positive then the first derivative is increasing- it starts low on the left so you have almost a horizontal line then increases so it becomes steeper and steeper. That gives you a "concave upward" graph. If the second derivative is negative, it is just the opposite- the graph starts upward sharply, then starts to level off- that is "concave downward".

So say I was to put a random x value into the second derivative and it spat out 24. Then, I put in a different random x value and it spat out say...26. What would the difference be between the two points?

## 1. What is the definition of a second derivative?

The second derivative of a function is the derivative of the first derivative. In other words, it is the rate of change of the rate of change of the original function.

## 2. How does the second derivative relate to concavity?

The second derivative tells us whether the slope of a curve is increasing or decreasing. If the second derivative is positive, the curve is concave up, meaning it is curving upwards. If the second derivative is negative, the curve is concave down, meaning it is curving downwards.

## 3. Why is the second derivative used to find points of inflection?

A point of inflection is a point where the concavity of a curve changes. This means that the second derivative is equal to 0 at this point. By finding the points where the second derivative is equal to 0, we can identify points of inflection.

## 4. Can the second derivative be used to find the direction of a curve?

Yes, the second derivative can also help determine the direction of a curve. If the second derivative is positive, the curve is increasing in a positive direction (to the right). If the second derivative is negative, the curve is decreasing in a negative direction (to the left).

## 5. How does the second derivative help us analyze the shape of a curve?

The second derivative provides information about the curvature of a curve. By analyzing the sign and magnitude of the second derivative, we can determine whether the curve is concave up or down, and identify points of inflection. This information helps us understand the overall shape and behavior of the curve.

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