Is f(x)=(-3x^3+15x^2-2x+1)/(x-4) increasing or decreasing at x=2?

1 Answer
May 25, 2016

decreasing at x = 2

Explanation:

To determine if a function 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 $\textcolor{b l u e}{\text{quotient rule}}$

If f(x)=g(x)/(h(x))" then" f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x)^2
$\text{-------------------------------------------------------------------}$

$g \left(x\right) = - 3 {x}^{3} + 15 {x}^{2} - 2 x + 1$

$\Rightarrow g ' \left(x\right) = - 9 {x}^{2} + 30 x - 2$

and $h \left(x\right) = x - 4 \Rightarrow h ' \left(x\right) = 1$
$\text{---------------------------------------------------------------}$
Substitute these values into f'(x)

$f ' \left(x\right) = \frac{\left(x - 4\right) \left(- 9 {x}^{2} + 30 x - 2\right) - \left(- 3 {x}^{3} + 15 {x}^{2} - 2 x + 1\right) .1}{x - 4} ^ 2$

$\Rightarrow f ' \left(2\right) = \frac{\left(- 2\right) \left(22\right) - \left(33\right)}{- 2} ^ 2 = - \frac{77}{4}$

Since f'(2) < 0 , then f(x) is decreasing at x = 2
graph{(-3x^3+15x^2-2x+1)/(x-4) [-29.65, 29.67, -14.86, 14.8]}