# Is f(x)=(-2x^3+4x^2-x-2)/(x+3) increasing or decreasing at x=-2?

Apr 26, 2016

decreasing at x = -2

#### Explanation:

To determine if a function is increasing/decreasing at x = a we evaluate f'(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

differentiate f(x) using the $\textcolor{b l u e}{\text{ quotient rule }}$

If $f \left(x\right) = \frac{g \left(x\right)}{h \left(x\right)} \text{ then } f ' \left(x\right) = \frac{h \left(x\right) . g ' \left(x\right) - g \left(x\right) . h ' \left(x\right)}{h \left(x\right)} ^ 2$
$\text{--------------------------------------------------------------}$

g(x) $- 2 {x}^{3} + 4 {x}^{2} - x - 2 \Rightarrow g ' \left(x\right) = - 6 {x}^{2} + 8 x - 1$

h(x) = x + 3 → h'(x) = 1
$\text{----------------------------------------------------------}$
Substitute these values into f'(x)

f'(x) $= \frac{\left(x + 3\right) \left(- 6 {x}^{2} + 8 x - 1\right) - \left(- 2 {x}^{3} + 4 {x}^{2} - x - 2\right) .1}{x + 3} ^ 2$

and f'(-2)$= \frac{1. \left(- 24 - 16 - 1\right) - \left(16 + 16 + 2 - 2\right)}{1}$

= (-41-32) =-73

Since f'(-2) < 0 then f(x) is decreasing at x = -2
graph{(-2x^3+4x^2-x-2)/(x+3) [-10, 10, -5, 5]}