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Determine all critical points for each function.$$f(x)=\frac{x^{2}}{x-2}$$

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$x=0$ and $x=4$

Calculus 1 / AB

Chapter 4

Applications of Derivatives

Section 1

Extreme Values of Functions

Derivatives

Campbell University

University of Nottingham

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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Find all critical points o…

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Determine all critical poi…

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Find the critical points o…

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Find all the critical poin…

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we are trying to find the critical point where this function first. We're going to recognize that X can't be too because that would mean in the function we're dividing by zero. Find the critical point. Let's examine the derivative derivative can be found using the quotient rule bottom time. So you wrote at the top minus the top Times The derivative, the bottom which is one over the bottom squared. So we have two X squared minus four, X minus X squared Over X -2, quantity squared. And that would be x squared minus four, X Over X -2. Right, quantity square. Now this is a non differentiable function at X equals two. But remember that is not the domain. So the critical points are critical numbers are going to come when our denominators are derivative is zero and derivatives are zero or fractions or zero When the top of the fraction is zero. So we will set X squared minus four, X equal to zero. Factoring out the X. We have X times X minus four is zero. And that would be zero if X zero Or when X -4 is zero, which is when X is four. So are critical points or one X is zero Or when X is four. The function value enact zero is zero over negative two or zero and effects is four. That'd be 16/2, which is a.

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