First we need to find the dots, because these dots will tell us the intervals we need to work with. Actually, for this part we only need to find the *x*-values of the critical points and possible inflection points. The derivative is *f *'(*x*) = -12*x*^{2} + 3 which is zero when We have critical points at . The second derivative is *f *"(*x*) = -24*x* which is 0 when *x* = 0, so this is a possible inflection point. Now that we've found the CPs and possible IPs, we make a numberline. The CPs and possible IPs divide the numberline up into intervals. We need to know the signs of *f *' and *f *" on each of these intervals. We draw dashed lines so we can keep the intervals straight: The derivative is positive when 12*x*^{2 }< 3 which occurs when The derivative is negative otherwise. Fill this in on the numberline: The second derivative is negative when *x* is positive, and positive when *x* is negative: Now we can see what shape *f* should have on each interval: The numberlines in the graphing exercises will also include a numberline for the sign of the function *f*, because seeing where *f* is positive and negative is useful for drawing the graph correctly. However, you don't need to know the sign of *f* to determine the shape of *f*. |