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Classify all critical points.$r(x)=4 x^{3 / 4}+2$

$$\mathrm{m}(0,2)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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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Locate all critical points…

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Classify all critical poin…

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01:30

Find all critical points x…

05:05

01:53

Find all the critical poin…

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Determine all critical poi…

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02:33

we want to find the derivative of the function are of X. So we do that by taking the derivative of each term. So the derivative of four X. 23 quarters is You bring this down here and multiply it by the force that gives us three X. and then subtract one from the exponents, the -14. And then the derivative of two since it's a constant is zero. So this is our derivative function. What we want to do next is set this equal to zero. So three X -5. Or to the power of -1 4th equal zero. Now we saw for X considering the fact that this is mm multiplication as in three times X. To the power of something X will equals or the function will equal zero when X equals zero. So that is our one critical point. And next what we want to do is find out the why Or that's actually named R X. When X equals zero. So you plug in zero to your original equation here. And when you solve for our backs you get y equals two Or are x equals two. So what we want to do next to figure out what kind of a criminal code zero is is we're going to plot zero and then pick a value slightly above and slightly below it. So we will pick negative one and 1. So at zero the derivative are of X. is zero. So looking at are a derivative again at -1. Our equation three X to the -1 4th will be negative. And then at and you don't actually have to solve it all the way through. Um You can if you want I just did it by like plugging in a negative. If I had a negative value for X. Would that give me a negative or positive value for the for the derivative? And then for one seeing as it's a positive number. Or we put him plug in 12 x. Um multiplied by three. Okay Take the derivative or sorry increased it to the expletive -14th. We get a positive value. So because it's Negative as in the derivative is decreasing on one side and then positive as in it's increasing on the other side on either sides of our critical value. We know that this is a minimum so X equals the point at the point 02 The Or at the critical .02. We have a minimum value for this function.

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