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Classify all critical points.$f(x)=2 x^{4}+2 x^{3}-x^{2}-7$

$$\mathrm{M}(0,-7), \mathrm{m}_{1}(1 / 4,-899 / 128), \mathrm{m}_{2}(-1,-8)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Missouri State University

University of Michigan - Ann Arbor

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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Classify all critical poin…

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Locate all critical points…

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Determine all critical poi…

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Find all the critical poin…

do you want to find the derivative of this function? We use it during power role By eight X. Cubed Because six x squared minus two X. And we can actually funk factor out of two here to leave us with we're sorry to X. To leave was us with four X squared plus three facts minus what? So you want to factor out we want to try to factor this binomial. So we can we have we can use negative one and positive for to get three right here. So we got four x squared last four X minus eggs minus one after that out. Okay. And then we so simplifying our Whole derivative. We got two x times four X. And it's one. Remember we got the two eggs from here Works -1 Year And then x plus one equals are derivative. Which we want to set equal to zero. So solving each of these expressions for zero we get our zeros to be our X values zero when X equals zero negative one and 1/4. So this part you can do on your own, plug in each of these X values to your original function to find the corresponding why values. And when you do that we get negative seven negative eight. And for 1 4th You get 8 99 over 1 28. Because this will allow you to have your ordered pairs your X and your Y. Values of when your function is our of your critical points. Now to find out what critical points we have, we're going to have a number line with your zeros and value slightly smaller and slightly bigger than your zeros. So we have 0, 1 4th negative one. And we're trying to find if these are minimum maximum or inflection points. So we know the derivative at each of these points is equal to zero. And so picking value slightly bigger and slightly smaller. Let's do a native to negative one half 18 and one. So we're going to just look at the derivative ah X equals each of these values. And we want to just see if it's positive or negative. So plugging in negative two to this equation will give us a well give us a negative derivative because we got negative here, negative here, negative here And then plugging negative 1/2 into the equation. Well, again, doing the same thing we did for the top one, we get a positive value very derivative 18 will give us a negative value and one will give us a positive value. So looking at your positives and negatives, you can tell that here you go from a negative derivative or a negative slope to a positive slope, which means it is a minimum. And then we get a maximum and then a maximum. Looking at the way that your slopes are reflected.

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