# Is f(x)=(-7x^3-x^2-2x+2)/(x^2+3x) increasing or decreasing at x=1?

Feb 21, 2018

It is decreasing at $x = 1$

#### Explanation:

Whether a function $f \left(x\right)$ is increasing or decreasing depends on value of $f ' \left(x\right)$ at that point. If $f ' \left(x\right) > 0$ i.e. it is positive, it is increasing and if $f ' \left(x\right) < 0$ i.e. it is negative, it is decreasing.

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

and using quotient formula, we have

$f ' \left(x\right) = \frac{\left({x}^{2} + 3 x\right) \left(- 21 {x}^{2} - 2 x - 2\right) - \left(- 7 {x}^{3} - {x}^{2} - 2 x + 2\right) \left(2 x + 3\right)}{{x}^{2} + 3 x} ^ 2$

and $f ' \left(1\right) = \frac{\left(1 + 3\right) \left(- 21 - 2 - 2\right) - \left(- 7 - 1 - 2 + 2\right) \left(2 + 3\right)}{1 + 3} ^ 2$

= $\frac{4 \cdot \left(- 25\right) - \left(- 8\right) \cdot 5}{16}$

= $\frac{- 100 + 40}{16} = - \frac{60}{16} = - 3.75$

Hence $f \left(x\right) = \frac{- 7 {x}^{3} - {x}^{2} - 2 x + 2}{{x}^{2} + 3 x}$ is decreasing at $x = 1$

graph{(-7x^3-x^2-2x+2)/(x^2+3x) [-4.58, 5.42, -4.1, 0.9]}