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Use the first derivative to determine where the given function is increasing and decreasing.$q(x)=x^{4}-32 x$ (Hint: $x^{3}-a^{3}=(x-a)\left(x^{2}+a x+a^{2}\right)$.)

Dec on $x < 2 ;$ inc on $x > 2$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

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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Use the first derivative t…

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So first we want to find the derivative of function Q. Of X. And we can use this do this using power rule. So for our first term X to the fourth we get our derivative to equal for X cubed. And then for a second term you get 32. Yeah. So using the relationship has given him the hint. We factor out the floor and we can see that we have a cubic variable minus a constant cubed. So in this case we can factor this relation dysfunctional To get X -2 times. That's weird. Plus two X. Plus four. So now we want to bring, make sure you remember this for over here. And so this is our derivative A function Q. So now we want to find where this function where This equation equals zero. And remember wearing a derivative equal zero. That's where the tangent is a straight line, like a horizontal line. So that means that it either you either have like a change in direction or it reaches a minimum or maximum and we'll see what that means in a little bit. So with each of these, if any of these equals zero, then our functional equals zero. Since it's the product of multiple uh multiple expressions so far does not equal zero. So we don't have to worry about that right now. Um X squared plus to expose for you can see it's not easily fact. Herbal. And even if you were to use the quadratic formula native B plus or minus the square root of He's Weird -4 a. c. All over to a. You can see that this right here would actually be a Square root of a negative number, which is not rational. So this expression can't be factored, which leaves us with X -2 -2 equals zero. And when we saw that at x equals to the whole derivative equals zero. So that is our zero. And what we can do to find out where this function is increasing or decreasing, we have to hear. And we know we can either plug that into our original equation and find the value and then compare it with the numbers slightly higher Or lower. So we'll see one or 3. Um or what we can do is we already plugged to into the derivative equation and we know that equals zero. So by plugging one and three into the derivative, we can see if the derivative is a positive or a negative value. If it's a positive value, that means that function is decreasing. If it's a negative value, it means it's decreasing. So at what we have a negative value because One to the Cube Times 4 -32 gives us a negative number and then at three we have a positive value. So that gives us our answer. Of the function is increasing uh X greater mhm than to and decreasing X is less than nope switched up my signs, so so decreasing X is less than two, and increasing X is greater than two.

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