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Find the local maximum and minimum values of $f$ using both the First and Second Derivative Tests. Which method do you prefer?$f(x)=\frac{x^{2}}{x-1}$

local max: $(0,0) \quad$ local min: $(2,4)$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

Campbell 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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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for this program That's first right off the first of the curative and the second of narrative. I'm so I want to find a local momentum and low commitment. It's a method one we're gonna use the first of the year. It here suggest Let's if Prime equals zero. So with two solutions, X equals zero and X equals 22 So we have three intervals to kids either for all minus um, be careful about this function because that domain off this function, the original function isse from negative infinity to one in the Union Union with 12 infinity. We on the function is no defined. It's no defined it when X equals toe. So in fact, we have four different intervals to can see there. So the first intervals from minus infinity to I am zero. So over this interval and for also the curative, it's negative. How sorry is positive. So F a C increasing, and then the second numbers from 0 to 1, 10 to 1. The first of the narrative is Mac there, so it's decreasing the 30 in her voice from 1 to 2 again. Force other. Do it if it's connective So it's the equation and the front two to infinity, the first of the purity. If it's positive. So it's increasing. Now we can crew that we have. So we have a local maximum here at X equals zero with with a value of zero, because there's zero. So this is a local maximum. We don't have anything for this. X equals to one, so there is no isn't. It's not. It's a critical point, but it's not a local maximum on no minimum, so we don't have sent. We don't have the result for this. X equals one, but we have resulted for X equals due to it's a low, cold millemann away with value as two equals toe for Okay, so this is This result comes for on the first of the narrative test, and then we can use the second the theater test to characterize the local maximum no convenient for critical points. Um, so we have to a great your points. So for for X, because 20 he packed X equals there into a second figurative. So we have to over minus one, so it's negative. That means he's a long hole mix for XY question to the second maturity because the tour over one which pointed so it's a local minimum, the same as the result we just got.

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