# Concavity of a rational function

1. Jun 29, 2011

### realism877

I have to curve sketch this function, ((x)3))/((x)2-1)

I did all of the steps, and I got this as the second derivative: ((2x(x2+3))/((x2-1)3)

I got concave up:(-1, 0)u(1, inifinity)

Concave down:(-infinity, -1)u(0,1)

Am I right?

I use -1 and 1 as the interval since they are the asymptotes.

2. Jun 29, 2011

### micromass

Staff Emeritus
Hi realism877!

That is correct.

3. Jun 29, 2011

### SammyS

Staff Emeritus
To have a more complete sketch, how does the function behave for |x| → ∞ ?

Where are inflection points and relative extrema?

4. Jun 29, 2011

### realism877

the point of inflection is (0,0)

5. Jun 29, 2011

### SammyS

Staff Emeritus
To see how the function behaves for |x| → ∞, it might help to write $\displaystyle \frac{x^3}{x^2-1}\ \ \text{ as }\ \ x+\frac{x}{x^2-1}$

6. Jul 3, 2011

### realism877

The local max=(-sqr(3), -2.598)

Local min=(sqr(3), 2.598)

Correct?