# How do you determine the end behavior of f(x)=1/3x^3+5x?

Feb 17, 2017

$- \infty$ to the left and $\infty$ to the right

#### Explanation:

to know the end behavior of a function you need to know two things:
$1.$ if the function is positive or negative
$2.$ what the base function is, and what the base function looks like

$f \left(x\right) = \frac{1}{3} {x}^{3} + 5 x$
$1.$ the function is positive, so it will be increasing
$2.$the base function is $y = {x}^{3}$

$g r a p h : y = {x}^{3}$
graph{y=x^3 [-10, 10, -5, 5]}
so knowing that $f \left(x\right) = \frac{1}{3} {x}^{3} + 5 x$ is positive and base function is $y = {x}^{3}$, it will be heading towards $- \infty$ to the left and $\infty$ to the right

the $\frac{1}{3}$ and $5 x$ just makes the function become a really thin $y = {x}^{3}$

$g r a p h : y = \frac{1}{3} {x}^{3} + 5 x$ (if you scroll out you'll see it better)
graph{y=1/3 x^3 + 5x [-10, 10, -5, 5]}