# Is f(x)=(-x^3-7x^2-x+2)/(x-2) increasing or decreasing at x=3?

Sep 21, 2017

$f \left(x\right)$ increases at $x = 3$

#### Explanation:

To find out whether a function is increasing or decreasing at one point, find the derivative of the function and determine if the derivative is positive or negative at that point.

First, differentiate $f \left(x\right)$

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \frac{- {x}^{3} - 7 {x}^{2} - x + 2}{x - 2}$

$= \frac{d}{\mathrm{dx}} \left(- {x}^{3} - 7 {x}^{2} - x + 2\right) \cdot \frac{1}{x - 2}$

$= \frac{d}{\mathrm{dx}} \left(- {x}^{3} - 7 {x}^{2} - x + 2\right) \cdot {\left(x - 2\right)}^{-} 1$

$= \frac{1}{x - 2} \cdot \frac{d}{\mathrm{dx}} \left(- {x}^{3} - 7 {x}^{2} - x + 2\right) + \left(- {x}^{3} - 7 {x}^{2} - x + 2\right) \cdot \frac{d}{\mathrm{dx}} {\left(x - 2\right)}^{-} 1$

$= \frac{1}{x - 2} \cdot \left(- 3 {x}^{2} - 14 x - 1\right) + \left(- {x}^{3} - 7 {x}^{2} - x + 2\right) \cdot - {\left(x - 2\right)}^{-} 2 \cdot \frac{d}{\mathrm{dx}} \left(x - 2\right)$

$= \frac{- 3 {x}^{2} - 14 x - 1}{x - 2} + \left(- {x}^{3} - 7 {x}^{2} - x + 2\right) \cdot - {\left(x - 2\right)}^{-} 2$

Simplifying,

$= \frac{- 3 {x}^{2} - 14 x - 1}{x - 2} - \frac{- {x}^{3} - 7 {x}^{2} - x + 2}{x - 2} ^ 2$

$= \frac{\left(- 3 {x}^{2} - 14 x - 1\right) \left(x - 2\right)}{x - 2} ^ 2 - \frac{- {x}^{3} - 7 {x}^{2} - x + 2}{x - 2} ^ 2$

$= \frac{- 3 {x}^{3} - 8 {x}^{2} + 27 x + 2 - \left(- {x}^{3} - 7 {x}^{2} - x + 2\right)}{x - 2} ^ 2$

$= \frac{- 3 {x}^{3} - 8 {x}^{2} + 27 x + 2 + {x}^{3} + 7 {x}^{2} + x - 2}{x - 2} ^ 2$

$= \frac{- 2 {x}^{3} - {x}^{2} + 28 x}{x - 2} ^ 2$

So to find out if the function is increasing or decreasing at $x = 3$, find if $f ' \left(3\right)$ is positive or negative

$f ' \left(3\right) = \frac{- 2 {\left(3\right)}^{3} - {\left(3\right)}^{2} + 28 \left(3\right)}{\left(3\right) - 2} ^ 2 = 21$

Since $f ' \left(3\right) > 0$, $f \left(x\right)$ has a positive slope at x=3, and thus $f \left(x\right)$ is increasing

graph{(y-(-x^3-7x^2-x+2)/(x-2))=0 [1.423, 4.492, -91.816, -90.28]}