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Use the second derivative test to classify the critical points.$$f(x)=(x-1)^{2}(x+3)^{2}$$

$$\mathrm{m}(-3,101), \mathrm{M}(-2,108), \mathrm{m}(1,-27)$$

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

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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Use the second derivative …

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Classify the critical poin…

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Find the critical points a…

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question 43 would like you to use the second derivative test to classify the points. The critical points for F of X equals X minus one squared multiplied by X plus three squared. Um, To do so, we need to first find f prime of X, which we need to use the product rule. So in this case, you is X minus one squared. Your prime is too X minus one. The is X plus three squared and V prime is to expose three. Putting that into our equation here X minus one squared times two times X plus three plus expose three squared times two times x minus one Simplifying that you get f prime of X is equal to expand its one squared times to expose six well X plus three squared times two X minus two Uh, then setting this equal to zero. You would find that the zeros are at X equals one negative three and negative one. You can also grab this on a graphing calculator and find the zeros, uh, to help you a little bit with that. So those are going to be your three critical points. So plugging those into F of X negative three is that zero negative one is at 16 and one is at zero. Next, you need to test whether they are a maximum or minimum. So again, we need to use the um, you could use the product rule, or you can actually multiply everything out to make it a little simpler. So f double prime or f f prime of X is equal to x squared minus two x plus one times two x plus six plus X squared plus six x plus nine roles played by two x minus two Continuing to simplify by multiplying that out two x to the third plus two x squared minus 10 X plus six plus two x to the third plus 10 x squared plus six x minus 18. Then bringing it all together, you'd have four x to the third plus 12 X squared minus four X minus 12. Now we can just take the F double primer bags, which is 12 x squared plus 24 X minus four. Now testing all these critical points to see their con cavity F double prime of negative three is 32 so concave up, which means it's a minimum F double prime of negative one is negative. 16. So concave down a max and F double prime of one. 32 concave up. That is a minimum. So those are your answers then? For question 43 Yeah.

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