# Is f(x)=(-x^2+3x+2)/(x^2-1) increasing or decreasing at x=2?

Jun 20, 2016

$f \left(x\right)$ is decreasing at $x = 2$

#### Explanation:

Whether a function $f \left(x\right)$ is increasing or decreasing at, say $x = a$, is indicated by the value of $\frac{\mathrm{df}}{\mathrm{dx}}$ at $x = a$. If it is positive the function is increasing and if it is negative, the function is decreasing.

As $f \left(x\right) = \frac{- {x}^{2} + 3 x + 2}{{x}^{2} - 1}$, using quotient rule

$\frac{\mathrm{df}}{\mathrm{dx}} = \frac{\left(- 2 x + 3\right) \left({x}^{2} - 1\right) - 2 x \left(- {x}^{2} + 3 x + 2\right)}{{x}^{2} - 1} ^ 2$

= $\frac{- 2 {x}^{3} + 2 x + 3 {x}^{2} - 3 + 2 {x}^{3} - 6 {x}^{2} - 4 x}{{x}^{2} - 1} ^ 2$

= $\frac{- 3 {x}^{2} - 2 x - 3}{{x}^{2} - 1} ^ 2$

and at $x = 2$, we have $\frac{\mathrm{df}}{\mathrm{dx}} = \frac{- 3 \times {2}^{2} - 2 \times 2 - 3}{{2}^{2} - 1} ^ 2 = - \frac{19}{9}$

Hence as $\frac{\mathrm{df}}{\mathrm{dx}}$ is negative at $x = 2$, $f \left(x\right)$ is decreasing at $x = 2$

graph{(-x^2+3x+2)/(x^2-1) [-8.62, 11.38, -4.16, 5.84]}