Is f(x)=(-2x^3-x^2-5x+2)/(x+1) increasing or decreasing at x=0?

Mar 19, 2016

decreasing at x = 0

Explanation:

To test if a function is increasing / decreasing at x = a , require to check the sign of 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

Require to find f'(x)

differentiate using the $\textcolor{b l u e}{\text{ Quotient rule }}$

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

$g \left(x\right) = - 2 {x}^{3} - {x}^{2} - 5 x + 2 \Rightarrow g ' \left(x\right) = - 6 {x}^{2} - 2 x - 5$

h(x) = x+1 $\Rightarrow h ' \left(x\right) = 1$
$\text{------------------------------------------------------------------------}$
substitute these values into f'(x)

$\Rightarrow f ' \left(x\right) = \frac{\left(x + 1\right) . \left(- 6 {x}^{2} - 2 x - 5\right) - \left(- 2 {x}^{3} - {x}^{2} - 5 x + 2\right) .1}{x + 1} ^ 2$

and f'(0) = $\frac{1. \left(- 5\right) - 2.1}{1} = - 7$

since f'(0) < 0 , f(x) is decreasing at x = 0

Here is the graph of f(x)
graph{(-2x^3-x^2-5x+2)/(x+1) [-10, 10, -5, 5]}