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Find the points on the curve $y=x^{4}-8 x^{2}+3$ at which the tangent line is horizontal.

$$(-2,-13),(0,3),(2,-13)$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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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The underlying theme in this problem is that we're finding the values four X. I guess I should mention that way. So I don't know what C is. But if we have a horizontal tension, that means that the slope is equal to zero. So as you're looking at this problem, what I'm gonna do is take the derivative of extra the fourth minus eight X squared plus three and set the derivative equal to zero. That z what we gotta do. So what's the derivative Will bring the four in front. It's now to the third power because you subtract one from your exponents two times eight is 16 next to the first and set that equal to zero. So now you can do this factor out a four x you're left with X squared minus four eso When you use a zero probably property, you can say Okay, well, ex convict zero. And I don't care what anything else is because if x zero we're going to get zero or x squared can equal four because that makes this piece zero on anything. Times zero would be zero. That means X could be plus or minus two. But If you pay close attention to the directions, it's asking for the points on the curve. So what that means is we have to take these values we found and plug them back into the original problem. Now plugging in zero is pretty easy because you get zero minus zero plus three. So that means the ordered pair of 03 The Y coordinate is one answer. Now these other ones. They're a little bit more complicated because if you plug in to 248 16 to to the fourth Power 16, just do some scratch work here. Two squared would be four times eight would be 32 so or at negative 16. So far, plus three gives me negative 13 to negative. 13 is another point on the curve. Um, so let's do the same thing with negative, too. Well, negative to to the fourth Power. Still positive. 16. Negative two to the second power. Still positive, so it's still negative. 32 s o. You know, you can use a calculator to verify, but it's gonna be the same y value on that one as well. Eso these air your three points 032 negative 13 and native to native 13

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