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Show that a smooth odd function passing through the origin has an inflection point there.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

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Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

02:38

Prove that a cubic functio…

02:10

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01:07

Consider a general quadrat…

So for this one, all we need is a smooth odd function. We're just trying to show ah that would have an inflection point. Mhm. Right at the center. So here music, different color. Maybe indicate that there. Okay, So if we were to do that here, basically, what would happen is that? Well, we know that if we have an odd function on the right side, it increases or it could increase. But if it does increase on the right side, then it has symmetry to the origin here, which means that it must go decrease down in the opposite direction there. And if it does that, we could definitely see that it goes from concave down to give up right Pat zero. So therefore it has that here. And so we could show the other option that we have here with the non function is that if we have it like this, it's another way that we can visualize this here is with the opposite direction. So we see that we could have echo like this. It must I don't like that here. So this would be can't give up. And then that cock your town. Okay? We see. It has to have it at the center point here, right? Because of the fact that we have, again, as we're seeing here, there's symmetry to the origin. It must pass through the origin there and have an inflection point there. Okay, that's all.

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