The first thing we're going to do with this absolute value equation is we're going to make it in a piecewise equation. So we need to figure out when is the absolute value positive versus when is the absolute negative. When we factor X ^2 - 9, we get X + 3 * X - 3. So we know to the right of three it's positive, in between -3 and three it's negative, and to the left of -3 it's positive. So Y is going to equal two X ^2 -, 18 if X is less than or equal to -3, or if X is greater than or equal to three, Y is going to equal 18 - 2 X squared if we're in between -3 and three. Now the equal portion doesn't matter which part it's on. It could have been at the bottom. Traditionally, we leave the equal portion with the positive. So now taking our first derivative, we get 4X if X is less than or equal to -3 or X is greater than or equal to three, and we get -4 X if we're in between -3 and three, our second derivative is going to be 4 and -4 at X less than or equal to -3 or X greater than or equal to 3 and -3 less than X less than 3, respectively. So our questions are we have to figure out what the derivatives are doing so we can figure out increasing, decreasing in concavity. When we look at our first derivative, our first derivative is going to be if we think to the right of three, if we put in 456, we're going to get a positive number. At three, we're going to switch from positive to negative because we need to put it now into the bottom one. And -4 times the number in between zero and three is going to be negative. In between 0 and -3 and negative times, a negative is going to turn positive. And then to the left of -3 we've got to go up to that top equation again. And a positive times a negative is negative. So that's our first derivative. Our second derivative to the right of three is going to be +4 or a positive number in between -3 and three is going to be negative, and to the left of -3 is going to be positive. So we know our original graph is going to be decreasing but concave up, decreasing, concave up, then we're going to be increasing and concave down, Then we're going to do decreasing and still concave down, and finally increasing in concave up. So looking at this very rough graph, we know that we're going to have a local Max at zero and absolute mins at -3 and three. So when we stick in Y of -3 we get zero and Y of three we get zero. That's why they're both absolute men's as we got the same Y value out. If the Y values had been different, we would have taken the smallest Y value for the absolute men. The other one would have been a local men. Our Y of 0 gave us out 18. So that's a local Max. Because if we look at the original graph, we know it's going to keep going up. So the local mins are -3 zero and three zero. There is no absolute Max. The absolute mins are at negative 3030. There are no points of inflection. We have no points of inflection because it -3 and +3 are first and our second derivatives are undefined. If the 1st and 2nd are both 0 or both undefined, it's inconclusive. So those are not points of inflection. There were cusps or corners occur and we can't have points of inflection unless it's a smooth continuous curve. So the correct graph is going to be the one that goes decreasing concave up, increasing concave down, decreasing concave down, increasing concave up. And this case, this example, it was letter C.