# Is f(x)=(12x^2-16x-12)/(x+2) increasing or decreasing at x=5?

Sep 25, 2017

$\text{increasing at } x = 5$

#### Explanation:

$\text{to determine if f(x) is increasing at x = a evaluate}$
$f ' \left(x\right) \text{ at x = a}$

• " if "f'(a)>0" then f(x) is increasing at x = a"

• " if "f'(a)<0" then f(x) is decreasing at x = a"

$\text{differentiate using the "color(blue)"quotient rule}$

$\text{given "f(x)=(g(x))/(h(x))" then}$

$f ' \left(x\right) = \frac{h \left(x\right) g ' \left(x\right) - g \left(x\right) h ' \left(x\right)}{h \left(x\right)} ^ 2 \leftarrow \text{ quotient rule}$

$g \left(x\right) = 12 {x}^{2} - 16 x - 12 \Rightarrow g ' \left(x\right) = 24 x - 16$

$h \left(x\right) = x + 2 \Rightarrow h ' \left(x\right) = 1$

$\Rightarrow f ' \left(x\right) = \frac{\left(x + 2\right) \left(24 x - 16\right) - \left(12 {x}^{2} - 16 x - 12\right)}{x + 2} ^ 2$

$\Rightarrow f ' \left(5\right) = \frac{728 - 208}{49} > 0$

$\Rightarrow f \left(x\right) \text{ is increasing at x = 5}$