# Is f(x)=(-5x^3-x^2-3x-11)/(x-3) increasing or decreasing at x=2?

Mar 6, 2017

$\text{increasing at } x = 2$

#### Explanation:

To determine if f(x) is increasing /decreasing at x = a, 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 $\textcolor{b l u e}{\text{quotient rule}}$

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

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{f ' \left(x\right) = \frac{h \left(x\right) g ' \left(x\right) - g \left(x\right) h ' \left(x\right)}{h \left(x\right)} ^ 2} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{here } g \left(x\right) = - 5 {x}^{3} - {x}^{2} - 3 x - 11$

$\Rightarrow g ' \left(x\right) = - 15 {x}^{2} - 2 x - 3$

$\text{and } h \left(x\right) = x - 3 \Rightarrow h ' \left(x\right) = 1$

$\Rightarrow f ' \left(x\right) = \frac{\left(x - 3\right) \left(- 15 {x}^{2} - 2 x - 3\right) - \left(- 5 {x}^{3} - {x}^{2} - 3 x - 11\right) \left(1\right)}{x - 3} ^ 2$

$\Rightarrow f ' \left(2\right) = \frac{\left(- 1\right) \left(- 67\right) - \left(- 61\right)}{1} = 128$

Since f'(2) > 0, then f(x) is increasing at x = 2
graph{(-5x^3-x^2-3x-11)/(x-3) [-10, 10, -5, 5]}