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

Apr 11, 2018

$f \left(x\right) = \left(x - 2\right) \left(x + 5\right) \left(x + 2\right)$ is decreasing at $x = - 3$

#### Explanation:

We are given $f \left(x\right) = \left(x - 2\right) \left(x + 5\right) \left(x + 2\right)$

Now the function is increasing at a given point, if $\frac{\mathrm{df}}{\mathrm{dx}} > 0$ at that point and is decreasing if $\frac{\mathrm{df}}{\mathrm{dx}} < 0$.

Here $f \left(x\right) = \left(x - 2\right) \left(x + 5\right) \left(x + 2\right) = \left(x + 5\right) \left({x}^{2} - 4\right) = {x}^{3} + 5 {x}^{2} - 4 x - 20$

and $\frac{\mathrm{df}}{\mathrm{dx}} = 3 {x}^{2} + 10 x - 4$

at $x = - 3$, we have ${\left(\frac{\mathrm{df}}{\mathrm{dx}}\right)}_{x = - 3} = 3 {\left(- 3\right)}^{2} + 10 \left(- 3\right) - 4$

= $27 - 30 - 4 = - 7$

Hence $f \left(x\right) = \left(x - 2\right) \left(x + 5\right) \left(x + 2\right)$ is decreasing at $x = - 3$

(see graph not drawn to scale - shrunk alon $y$-axis)
graph{(x-2)(x+5)(x+2) [-10, 10, -40, 40]}