function has either a vertical tangent or a cusp at x = 0. We’ll be able to determine which after
we take the second derivative. Notice also that the derivative is negative for x < 0 and positive
for x > 0. Therefore, the curve is decreasing for x < 0 and increasing for x > 0. Next, we take
the second derivative: . If we set this equal to zero, there is no solution. The
second derivative is always negative, which tells us that the curve is always concave down
and that the curve has a cusp at x = 0. Note that if it had switched concavity there, then x = 0
would be a vertical tangent. Now, we can draw the curve. It looks like the following: