 Hi there and welcome to this screencast where we're going to work with a family of functions to determine some of the behaviors of this family. Specifically we're given the family of functions f of x equals a times the square root of b minus x squared, where a and b are positive real numbers. And for every member of that family, I would like to determine the domain, the critical values, the increasing, decreasing behavior, concavity and inflection points of this family. So first of all realize that although it says f of x, this is not one function, but it's an entire collection or family of functions that all have the same basic form. The only difference here is the presence of two parameters, a and b. Remember, very importantly for what's about to happen here is that a and b are not variables, they are numbers, they are positive numbers. We lock in values of those parameters and then use x as the variable. But in terms of derivative taking and everything else, a and b are numbers whose values we don't know. So let's start with the domain of this family. Now, the domain of this function here is going to include all x values that make the function computable, which in this case means I need all x values that make the expression that's under the radical sign zero or positive. I don't want any negative values under the radical sign. So I need to have b minus x squared to be bigger than or equal to zero. That's not the answer, I need to get the answer in terms of x. But just realize that the domain depends upon b here. Let's keep working with this and solve this inequality for x. I want b minus x squared to be non-negative, either zero or bigger than zero because of the radical sign there. So solving that inequality means if I add x squared to both sides, I get b bigger than or equal to x squared, which it seems a little bit more comfortable to write x squared less than or equal to b. So to solve this inequality, I'd want to take the square root of both sides here. But this is an inequality, so when I take the square root of both sides, I get not x less than or equal to radical b. But the absolute value of x less than or equal to radical b. And so if I solve that inequality in turn, I'm going to get negative square root of b less than or equal to x less than or equal to positive square root of b. That's a good inequality way of expressing the domain. If I want to put it in interval format, my domain would be the interval starting at negative radical b and extending to positive radical b and including those two end points. Now that we have the domain of my family of functions, let's move on to talk about the critical values and the increasing, decreasing behavior of this family. That's going to involve the first derivative. Now remember, when taking the derivative a and b are constants, they are numbers. They're just to be treated like regular numbers, even though we don't know what their values are. And I say that because we need to take the first derivative now. The first derivative of f would be if I take the derivative of a times radical b minus x, a is a number, it's a constant. So I can pull it out from the derivative process. And instead, I'll take the derivative of just the radical portion, which I'm going to write as b minus x squared to the one half power. Now if I take the derivative of that expression, that's going to involve the chain rule. Let's work this out. That's going to give me a, because that constant factor is still there, times one half times b minus x squared to the minus one half. And then times the derivative of the inside, which is negative 2x. Just to simplify the one half and the two cancel off, there's still the negative sign there though. So let's put this together. This is negative a times x times b minus x squared to the negative one half. Okay, and I'm going to do one extra step and that is to take that b minus x squared to the negative one half power and simply write it on the bottom of a fraction. So this is going to be negative ax over square root of b minus x squared. So that's in simplified form. Now the reason I'm doing this is because the problem was asking me to find the critical values of the family and also the increasing decreasing behavior of this family. Now the critical values of my family f are going to be the places as we know where the derivative, first derivative, is either zero or undefined. So let's think about those two conditions in turn. Where would f prime of x be equal to zero? Well, if f prime of x were equal to zero, the only way a fraction can be equal to zero is if its numerator is equal to zero. Now a is a positive number, so that can't be zero. So the only place where this fraction f prime can be zero is where x is equal to zero. So x equals zero is a critical value for every member of this family. Unlike the domain we saw, which does depend upon the parameter value b, notice that the critical value does not. Every single member of this family has a critical value in the exact same place at x equals zero. Now we also need to think about where the derivative is undefined. There actually are a couple of places where this is undefined now. Namely where the denominator of this fraction is equal to zero. The denominator of the fraction is equal to zero at x equals plus and minus radical b. Those are the very, very right and left end points of the domain. You put those things in for x squared and you end up dividing by zero. So there are three critical values for this function zero and plus or minus b. Now the next thing we need to think about is the increasing decreasing behavior of this function and I'm going to go to a different page to do that. So finding the increasing and decreasing behavior of really any function, including families of functions, involves knowing the sign of the first derivative. And that's helped along by making a sign chart as we've seen before. Here's my sign chart. Now notice this sign chart does not extend forever off to the left and to the right because the domain of the function does not extend off forever to the left and the right. The domain starts on the left at negative square root of b and ends on the right at positive square root of b. So when you visualize it like this, there's really only two intervals to check. This interval from negative radical b up to zero and then the other half of the interval from zero to positive square root of b. Just like before, I'm going to find the sign of f prime on each of those two intervals and then will directly give me the increasing decreasing behavior of f. Now this f prime is not a product of several things, like we've seen before, it's a quotient of two things. We can still think of the sign chart the same way. Especially when you notice that the denominator of the fraction in f prime is always positive. The denominator is a square root and that's never going to output negative results. So really we only need to look at the numerator of f prime. In this particular case, the only way f prime can be positive is for the numerator to be positive and vice versa. So let's look at values from each of the two intervals that we have here, created by the critical value at x equals zero. In the left-handed interval over here for values between negative radical b and zero, those are all negative values. And so if I put a negative value in for x, I'm taking x, which is negative, times a, which is positive. And then there's a negative sign out here. So really I have a negative value here times another negative value times that positive a, and that gives me a positive value for f prime. Now likewise on the other side here in this area, I'm looking at positive values of x. These are values of x that sit between zero and square root of b. So I have a negative sign here. A is always positive and x in this case is positive. So that's a positive x times a positive a and then I take a negative sign. That's going to give me a negative value for f prime. Now let's check out the behavior of f which is the punch line for this particular part. Since on the first interval, the left-handed interval, f prime was positive. That means f is increasing on that interval. And on the right-handed interval, f prime is negative. So f is decreasing on that interval. So what this tells me is I can now answer the question about where is the family of functions increasing and where is it decreasing. Well, I know now that f is increasing. Again, this is for any value of a and b. f is increasing on the interval from negative square root of b to zero and decreasing elsewhere, decreasing on the interval from zero to square root of b. Notice that this also gives us some other information for free that we weren't asked for. Namely, that is every member of this family is going to have a local maximum at x equals zero. No, we might ask what is the value of that local maximum? Let's do that right here. So this family, every family member is going to have a local maximum at x equals zero. And the local maximum value is f of zero, which is a times b minus square root of b minus zero squared, which is a radical b. So there's a little bit of free information. Every member of the family has a local maximum value, actually an absolute maximum value if you look at it, of a radical b. And that happens at x equals zero. Now we're going to move on to look at concavity. And this is a fairly lengthy computation. So we're going to do this in a separate screencast. So stay tuned for the sequel.