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Determine where the function is concave upward and downward, and list all inflection points.$$f(x)=x^{4}-6 x^{2}+9 x-2$$

$$\mathrm{CD} \text { if }-1<x<1 ; \mathrm{CU} \text { if }|x|>1, \mathrm{I}(-1,-16) \text { and }(1,2)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Harvey Mudd College

Baylor 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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Determine where the functi…

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Find the intervals on whi…

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Determine where the graph …

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question 27 would like you to determine where the function F of X equals X to the fourth minus six X squared plus nine X minus two is concave up or down, uh, and list all inflection points. So to do so, you need the second derivative of F of X, so F prime of X is equal to four X cubed minus 12 X plus nine If the double prime of X men is equal to 12 x squared minus 12. Uh, to find inflection points, you need to find where F double prime of X is equal to zero, so zero equals 12. X squared minus 12 zero is equal to 12 times X squared, minus one. Therefore, X would be equal to negative one and one plugging in to F of X at negative. One half of negative one is equal to negative 16 and F of one is equal to two. So these are your two inflection points. Ah, then testing con cavity. We can test in intervals. So when X is less than negative wine testing a number in F double prime of X, so f double prime negative two would be 36 Therefore, you are concave up when X is less than negative one. When X is in between negative one and one. Testing a number after the prime of zero would be negative. 12. Therefore, when excess between negative one and one, your concave down and lastly, testing when X is greater than one F double prime of two is equal to 36. Therefore, when X is greater than one, you're a concave up and those are your answers then for question 27.

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