# Square root and rational function.

.

For this function to be defined, we need that and .Thus, this function is defined on .

The function is a rational function, thus it is differentiable on its domain; the composition of this function with the square root function is differentiable where the rational function does not vanish. Finally is the product of this composition with a linear function, hence is differentiable on . Let us compute the first derivative:

This first derivative vanishes if, and only if:

i.e.

 or

Only the first one belongs to the domain of . Now for in the domain of differentiability, we have:

 or

The function increases on and on , and it decreases on .

Let us compute the limits of at the open endpoints of its domain:

Therefore the line whose equation is is an asymptote to the graph of .

We shall now look for oblique asymptotes:

Now we can show that

This means that the line whose equation is is an oblique asymptote to the graph of . Actually, the same line is an asymptote to for in a neighborhood of (please check this!).

We can now draw the variation table of (where and ):

Let us look for a point of inflexion. By arguments similar to those we used for differentiating , we prove that is differentiable on and we have:

The second derivative vanishes at 2, it is negative for and positive for ; thus, is concave on and on and it is convex on . At 2 the function has a point of inflexion.

Noah Dana-Picard 2007-12-28