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Find a cubic function $f(x)=a x^{3}+b x^{2}+c x+d$ that has a local maximum value of 3 at $x=-2$ and a local minimum value of 0 at $x=1.$

$f(x)=\frac{2}{9} x^{3}+\frac{1}{3} x^{2}-\frac{4}{3} x+\frac{7}{9}$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

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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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Find a cubic function $f(x…

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Suppose that $f(x)=A x^{2}…

in this question we have. We have four. Alan Determined coefficient is easy, Andy, and we want to figure out what's lay. But we have some informations. So this first take the curative. So we have if prime, because the three a x script pass to the X process, See? And we know if crime of minus two recourse to have prime of one Yukos zero so we can set up a system of equation. How we choose, probably minus to unwind 12 prime and the seven to be zero since 12 a minus four p press seek was zero in them. Three a process to be process See, cause zero So this to our 1st 2 equations we have. But we also know we also known that f minus two because of 3.5 way because zero which gives us another two equations. So a press B plus c pressed equals Teoh zero and the miners takes a press for P minus two C. Presti because the three so we're gonna solve this ceased all the questions that we can get. What a p c and the So the strategy to soft this, um equation is to first. Look it the last two. So, uh, this label obvious like you question 1234 saying Look it three and a four You question the rent a fork we use equation three miners equation for So we have 90 a minus three p press The receive equals zero and we divide this by three. So we have three a minus. B plus c equals zero and the re label this as equation five. Okay, so with with the equation 12 and five, we are able to self for a be here by eliminating. See? So for example, we use ah, one question one minus the question too. So we get night eight minus six. B equals 20 and that we use equation two miners. See question three. So we have, um, three D. So I made a mistake here. So when we used Equation three subject equation for we have this equals to minors. Mine is three. That means this is my this one. Okay, so when will you see? Question told, minus the equation. There we have three p equals to one. Therefore our first perimeter. Because toe 1/3 and re project back to get a a close to toe over. Then we problem back again to either one equation. White Christian Torrey. Question five. So we have CE close Teoh minus four or three. And the party off this three back to the question three. We have P equals to seven dominant.

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