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Show, by means of a sketch, that if a continuous function with a single critical point that is a relative extremum, then this critical point is also an extremum.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 1

Extrema of a Function

Derivatives

Missouri State University

Campbell University

Harvey Mudd College

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.

01:33

Sketch the graph of a func…

06:14

Show that if $f^{\prime \p…

02:20

Suppose that $c$ is a crit…

01:05

Draw a graph of a continuo…

00:15

Use the given derivative t…

01:25

Suppose that $f$ is contin…

00:28

Sketch a continuous functi…

we have to sketch the graph of a function which has a critical point. That is also an inflection point. So this question is challenging our graphing abilities specifically. It's also challenging our understanding of what critical point and inflection points are as such. Let's define what critical points and inflection points are visually. So we're able to graft correctly. So remember that a critical point on the top occurs when X goes from increasing to decreasing or decreasing the increasing. That is the sign of that crime changes. An inflection point, however, is when the F double prime changes. It's when the con cavity of F changes. Hence the two points I've marked here vest a graph this okay, we need to have that this act changes from increasing, decreasing or increasing decreasing and has this changing from cavity. So as an example, we can produce the function below. So a graph here has the necessary criterion. We see that at our point we go from increasing decreasing and we have a sharp change from concrete up to concrete down, or rather from concrete down to content. Up. This, we see is concrete down this week. You can't give up. Thus, we have a clear ascend tote at our point, but continuous, it satisfies the requirements of this problem.

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