.

This function is defined on .

The function is the product of the natural logarithm by a monomial, hence it is differentiable on its domain. We have:

Let us study the sign of the first derivative:

or or | ||

or |

Recall that is defined over , thus the first derivative vanishes only at ; it is negative over and positive over . The function decreases over and increases over . It has a minimum at .

Now let us check the limits of at the open ends of its domain.

Does the graph of the function have an oblique asymptote:

Thus there is no oblique asymptote.

Now, let us check how the graph looks like near the origin: We have

therefore:

Now let us compute the second derivative:

We have:

Thus is concave over and convex over . It has a point of inflexion at .

The graph of is displayed in Figure 3.

Noah Dana-Picard 2007-12-28