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Determine where the function is concave upward and downward, and list all inflection points.$$f(x)=(x-1) /(x+1)$$

$$\mathrm{CU} \text { if } x<-1, \mathrm{CD} \text { if } x>-1$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Oregon State University

Harvey Mudd College

Idaho State University

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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Determine where the functi…

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Determine the intervals on…

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question 29. Um is asking you to determine where the function F of X equals X minus one over X plus one is concave up or down and list inflection points. Uh, To do so, we need to find the second derivative of F of X using the quotient rule. Here we can find F prime of X, which is equal to X plus one minus X minus one, divided by X plus one squared simplifying that that would just be too over X plus one squared, which is equivalent to two times X plus one to the negative, too. Now taking the derivative again to get F double prime of X, that would be negative four times X plus one the negative three. So negative four over X plus one Cute, um, in this case F double prime of X Uh, when it's equal to zero, that would be an inflection point. This function is never equal to zero, but there is an ASEM tote at X equals negative one. Um so to find if it's concave up or down, we can just test that like an inflection point. So when X is less than negative one test F double prime of something like negative two. That's equal to four, which means when X is less than negative. One your con cave up and testing when access greater the negative one. Something like F double prime of zero that gives you negative four. Which means when X is greater than negative one, your concave down and those are your answers for Question 29.

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