# How do you use the first and second derivatives to sketch  f(x)=(x+2)/(x-3)?

##### 1 Answer
Jan 8, 2017

graph{(x+2)/(x-3) [-15.58, 24.42, -8.16, 11.84]}

#### Explanation:

$f \left(x\right) = \frac{x + 2}{x - 3}$

$f ' \left(x\right) = \frac{x - 3 - x - 2}{{\left(x - 3\right)}^{2}} = - \frac{5}{{\left(x - 3\right)}^{2}}$

$f ' ' \left(x\right) = \frac{10}{{\left(x - 3\right)}^{3}}$

We can now analyse the behaviour of the derivatives to sketch the function:

(1) $f ' \left(x\right) < 0$ everywhere in its domain $\mathbb{R} - \left\{3\right\}$, so $f \left(x\right)$ is strictly decreasing and has no local extrema.

(2) for $x < 3$, $f ' ' \left(x\right) < 0$ so $f \left(x\right)$ is concave down in $\left(- \infty , 3\right)$

(3) for $x > 3$, $f ' ' \left(x\right) > 0$ so $f \left(x\right)$ is concave up in $\left(3 , + \infty\right)$