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Find a cubic function $ y = ax^3 + bx^2 + cx + d $ whose graph has horizontal tangent at the points (-2, 6) and (2, 0).

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04:16

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 1

Derivatives of Polynomials and Exponential Functions

Derivatives

Differentiation

Missouri State University

Campbell University

Harvey Mudd College

Baylor University

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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he had slower. So when you right here. So we have. Why is equal to a X cute? Must be X square plus feet x plus D you differentiate this to get three a X Square plus two B X? Let's see. We're gonna set up our equations from our given points, so we have negative to calm a six when we plug it into the original equation. We got six in our line ISS negative. Eat a That's four p minus two C plus T is equal to six. Then we have 0.2 comma zero. When we plug it into our original equation, we good cereal. Yeah, a Ame plus four B plus two C plus D is equal to zero. Where are derivatives since there's horizontal tangents, the derivative for positive and negative, too, is equal to zero, so we go for negative to calm a six. You get our derivative to be equal to zero well, a minus four B plus C. It's equal to zero and for our point to calm a zero. This is also there as well, since there are horizontal tangents. 12 a plus four B plus C is equal to zero then we're going to subtract our equations. 12 day was for P. Thus he is equal to zero minus 12. A finest for P plus C is equal to zero. Then we got B is equal to zero. We're gonna add up our equations. Our 1st 2 equations less for B plus two c rusty this equal to zero. Then we got Dee as equal to three. Since we plug in, zero for B C is equal to negative 12 A. When we get a was four times zero plus two negative 12 a plus three is equal to zero. Can we get a value of three over 16? And when we plug it in, C becomes negative nine over four. Then we just plug it in to get wise people to three over 16 x square x cubed Excuse me minus nine over four x plus three

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