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Consider the function defined by $f(x)=x^{4}+8 x^{3} .$ (a) For what values of $x$ is $f^{\prime}(x)>0 ?$ (b) For what values is $f^{\prime}(x)<0 ?$ (c) At which point(s) will the tangent line be horizontal?

(a) $-6<x<0$ or $0<x$(b) $x<-6$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

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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.

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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as we examine this function, you could use some rules from altar it to to really understand, Um, based off the fact that we have a positive leading coefficient and an even degree. I don't know how many times this function changes sizes. Um, but I do know the end behavior, so I know it's going to decrease first, and it's got an increase at the end. I don't know what those values are. Um, but what you could do is you can find the derivative should be for execute plus 24 x squared. Um, you know, following our rules of multiplying in front and subtracted one from your action and setting that equal to zero. So what you could do is factor out four x squared into the problem and you'd be left with X plus three? Nope. Four times three is 12 x plus six. And if you don't believe me, distribute that in there and you should get the same thing. So what that tells me is that ex ago zero has a multiplicity of to eso that's just gonna touch the access there, and then X equals negative six as a multiplicity of one. Sometimes you might even see a sign charge that looks something like this. You 600 because there's two zeros there, uh, negative 60 and signed charts of pretty long way of explaining that the function if we have three negatives, negative times, negative times, negative is a negative. So we're going to have a from negative infinity until negative six way have a deep. That's basically saying that F prime of X is less than zero because three names makes it negative. Whereas if you look here a positive times, negative times negative, it's positive. And then over here, positive times, positive terms positive is positive. S So what that means is from negative 6 to 0 and also from zero to infinity. This is interval notation, By the way, uh, the first derivative is positive now, as far as horizontal tangents go where we kind of already figured that out as well. Um, the tangent line will be horizontal at X equals zero. Next was, uh, negative six. So that's our answer to part C. I didn't realize I didn't that out of order, but anyway, you have A and B here. I forget which one saying Which ones be, um, that's how I would do this

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