Determine the largest open interval of the domain (a) over which ...

Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = (x + 3)^2

Accepted Answer

Hey, everyone in this problem, we're asked to consider the function F of X is equal to X plus eight all squared. And to find the largest open interval of the domain at which the function is increasing and decreasing, we're given four answer choices, options A through D that each have different intervals for the increasing and decreasing portion of this problem. Now, let's start by thinking about what our function looks like. We have F of X is equal to X plus eight squared. Now, this polynomial has a degree too. So we know that this is a parabola. Yeah. So we have degree two. This is a Parabola and we have a positive leading coefficient. OK. The leading term if we were to multiply this out would be X squared and that has a coefficient of 11 is a positive number. So we have a positive leading coefficient. OK. And recall that, that tells us that this opens upwards. Hm So we have our Parabola, it opens upwards if we think about the areas or the intervals where this is increasing and decreasing. Well, it's gonna switch right at the bottom and this point is actually the vertex of our problem. On the left hand side of that vertex, we can see that this function is decreasing, which we're going to show in red. OK. So this is where our function is decreasing. And on the right hand side of that vertex, we can see that our function is increasing, which will show in green. So in order to figure out the interval where it's decreasing and increasing, what we want to do is find this vertex, OK? Because that's the point where we're gonna switch intervals in order to find the vertex. Let's think about the vertex form of our parabola. OK. Recall that if we have Y is equal to a multiplied by X minus H squared plus K, then the vertex is given by H comma K. Now our function F of X is already in this format. OK? A is just equal to one H is equal to negative eight in K is equal to zero. OK. So the vertex of the para we were given is gonna be located at negative eight comma zero. Now let's relate this to our intervals. OK? We know that the X value where this vertex occurs is negative eight. OK. This is the point negative 80. So to the left of this vertex, the X values are going to be less than negative eight. OK. That means our decreasing interval. It's gonna go from negative infinity. OK? All the values on the left hand side all the way up to this vertex of negative eight. OK. We use round brackets to indicate that we're not including any of these end points. OK. Right. At that vertex, this function is actually not increasing or decreasing. All right. So we have our decreasing interval. Let's do the same for our increasing interval. Now, this function is going to be increasing to the right of that vertex. OK? To the right means bigger X values. And so this interval is gonna go from negative eight all the way up to positive infinity. And again, we're not including those end points. All right. So we have our interval of decrease from negative infinity up to negative eight, we have our interval of increase from negative eight up to infinity. And this is gonna correspond with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = (x + 3)^2

Key Concepts

Function Behavior

Derivative

Critical Points

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