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Classify all critical points.$f(x)=x^{2}-2 x+3$

$$m(1,2)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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.

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Classify all critical poin…

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Locate all critical points…

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Determine all critical poi…

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This is Chapter three, Section two, Problem 15. And this problem gives us a function f of X and asked us to classify all of the critical points. And if you remember, critical points are where the derivative of a function is equal to zero. So first, let's start off finding that f prime of X, and that's just going to be equal to two X minus two. Yeah, so wherever two x minus two equals zero, that's going to be a critical point. Yeah, it's pretty easy to solve for X here. Looks like we get X equals one as a critical point. So we have our X and Y values. We know our X component of our critical point is one to find ry component. We just plug in that one into our original function up here. That's going to give us one squared minus two times one plus three. So that gives us a Y value of two. So are critical. Point is 12 and now we're trying to figure out whether that's a minimum or a maximum. So how we're going to do that is we are going to draw our number line here. We're going to find our critical point, which happens at X equals one at X equals one. We get zero. Yeah, we're going to pick an X value that's less than one, and we're going to pick an X value that's greater than one. Let's just choose X equals zero in X equals two. Yeah, we're going to plug both of these values into F Prime of X to see whether our function is increasing or decreasing. So let's start with X equals zero. If we have ex prime of X equals zero, we're going to have to time zero minus two. That gives us the value of negative two. And because it's negative, we know that if we have any values of X, less than one are function is going to be decreasing. What now? Let's try X equals two. Okay, we have f prime of two. We're gonna get two times two minus two. That's going to give us a positive, too, because it's positive. We know that any X values greater than one are going to result in an increasing function. So if we take this number line and we do a little sketch, yeah, we have our X equals one here and we know that zero. Yeah, we know that any X value that is less than one our function is going to be decreasing. Yeah, which means from here down. Yes, our function is decreasing. Okay. The opposite is true for an X value greater than one. We know that if we have an X value greater than one, our function is going to be increasing. So you can tell by our graph that we have a minima at one. Yeah. Two mhm.

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