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As in the preceding exercise, sketch the graph of the function passing through the relative maximum, $M,$ the relative minimum, $m$ and the inflection points$I_{1}, I_{2}, I_{3},$ and $I_{4},$ as given in Figure $27 .$ Assume that the derivative fails to exist at $M,$ but $f^{\prime}(x)$ and $f^{\prime \prime}(x)$ exist everywhere else.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Harvey Mudd College

University of Nottingham

Idaho State University

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.

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this one being asked to sketch what this would look like, got their first inflection point here and it's going to hit this minimum here. So for that to happen, we've got to be concave up Between inflection .1 and two. And so therefore, if it's can't give up there then behind inflection 0.1 before that would have to be car Cape Town. So we've got down to up and then it was going to switch going to From 2 to 3. So now the thing about him though is that this it's derivative does not exist. Mm So we have F prime. It's got to be DNA. So if that's the case that means but it could look like this. So you've got here got it concave down con keep up. And then from there it goes called Cape Town. And then so right, so that it would be ka cave up. Then I gave up again. And that call came down right going from So from my four. And so the reason why is because it said that the first derivative and secondary have existed everywhere else, except that the first route of was DNA for big. I'm here. Okay. And that's Telegraph should look like.

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