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(a) Draw the graph of a continuous function increasing and concave downward on $-1 \leq x<2$ and decreasing and concave upward on $2<x \leq 5$(b) Can such a function have any relative extrema or inflection points?(c) Suppose you have located the relative maximum, $M$, the relative minimun $m,$ and the inflection points $I_{1}, I_{2}, I_{3}, I_{4},$ and $I_{5} .$ Draw the smooth graph through these points. (See Figure 26 )

(a)(b) yes, it can have both.(c)

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Campbell University

Harvey Mudd College

Idaho State University

Boston College

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.

01:23

Sketch the graph of a fun…

01:00

Sketch the graph of a func…

01:01

Let $f$ be a continuous fu…

04:59

The graph of the derivativ…

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03:08

You are given $f^{\prime} …

So for this problem we're being asked to photograph of a continuous function. So first is increasing and concave downward. So I started a negative one increasing in conclave town look like that next part it's decreasing In concave upward. On this can look like that decreasing. Can't give up period here. Should we change that to period of five? Okay. They also said that the function was continuous. So this is what we have here and now for so you can sketch is any number of ways. As long as it's a uh huh like U. Shaped downward and as it's increasing and then over here it has to be a U. Shape up word for it to be can't give up and for decreasing there so you can make it a little more steep if you'd like or you know a little more flat. It's totally fine. Right? So for part B. They're asking if it can have any relative extra or inflection points. So the answer to that is yes. For both. the reason why is that too? So I text equals two. It's actually the relative max and it's also the an inflection point. There you go. So it's also an inflection point as well, that x equals two. Okay. So we have those. Alright now for part C. So they gave us a set of points and they want us to draw a smooth graph through them. So first part is what we have here connecting to I too. And so that see so I was out here. All right. So I fixed it up a little bit here. Uh That's in some dots here. So levelled out here changing con cavity so And so it's going through see here. Okay, So we start out here, this cocky of up then it's going to go concave down. So we connect these two here and then it will go concave up again and then switch the conclave down, hit this maximum, and then I'll go all the way here and then switch again to be Yeah, This one over here at I one so would have it be concave down could be like this, right? That's her answer.

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