# How do you describe the end behavior for f(x)=-x^4+x^2+2?

Jan 18, 2017

$f \in \left(- \infty , \frac{9}{4}\right]$. ]See explanation. and graph, for the overall behavior.

#### Explanation:

$f = - {x}^{4} \left(1 - \frac{1}{x} ^ 2 - \frac{2}{x} ^ 4\right) \to - \infty$, as x to +-oo.

$f = \left(2 - {x}^{2}\right) \left(1 + {x}^{2}\right) = 0$, at $x = \pm \sqrt{2}$.

$f ' = 2 x \left(1 - 2 {x}^{2}\right) = 0$, at $x = 0 \mathmr{and} \pm \frac{1}{\sqrt{2}}$

$f ' ' = 2 \left(1 - 6 {x}^{2}\right) = 0$, at $x = \pm \frac{1}{\sqrt{6}}$,

$\mathmr{and} > 0$, at turning points $\left(\pm \frac{1}{\sqrt{2}} , \frac{9}{4}\right)$,

giving global maximum $\frac{9}{4}$, at $x = \pm \frac{1}{\sqrt{2}}$-

At the turning point (0, 2), f''=2 giving local minimum 2.

$f ' ' ' \ne 0$, at x = +-1/sqrt 6, giving

POI , at $x = \pm \frac{1}{\sqrt{6}}$.

graph{-x^4+x^2+2 [-5, 5, -2.5, 2.5]}