# How do you describe the end behavior of f(x)=x^10-x^9+5x^8?

Jan 17, 2017

$f \ge 0$. As $x \to \pm \infty , f \to \infty$ See the graph.
The rate of rise of a function to $\infty \uparrow$, in this order :
$\ln x \left(x {x}^{2} {x}^{3} \ldots {x}^{10.} . .\right) {e}^{x}$

#### Explanation:

$f = {x}^{8} \left({x}^{2} - x + 5\right) = {x}^{8} \left({\left(x - \frac{1}{2}\right)}^{2} + \frac{19}{4}\right) \ge 0$

Also,

$f = {x}^{10} \left(1 - \frac{1}{x} + \frac{5}{x} ^ 2\right) \to \infty$, as $x \to \pm \infty$.

The rate of rise of a function to $\infty \uparrow$, in this order :

ln x (x x^2 x^3 ,,,x^10...) e^x

As the derivatives of f, up to the order 10, are 0 at x = 0, the graph is

horizontal, at the 10-tuple point O. I would like to call this 10 as order

of $f l a t i t u \mathrm{de}$.

Definition: If the derivatives of f are the same, up to and inclusive of order m at x = a, with ${f}^{\left(m + 1\right)} \left(a\right)$ different, the order of $f l a t i t u \mathrm{de}$ of the point x = a is m.

graph{x^8(x^2-x+5) [-5, 5, -2.5, 2.5]}