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

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Answer 1 · 2017-01-18T12:25:01

$f in(-oo, 9/4]$ . ]See explanation. and graph, for the overall behavior.

$f=-x^4(1-1/x^2-2/x^4) to -oo$ , as x to +-oo#.

$f=(2-x^2)(1+x^2)=0$ , at $x = +-sqrt2$ .

$f'=2x(1-2x^2)=0$ , at $x=0 and +-1/sqrt2$

$f''=2(1-6x^2)=0$ , at $x=+-1/sqrt6$ ,

$and > 0$ , at turning points $(+-1/sqrt2, 9/4)$ ,

giving global maximum $9/4$ , at $x = +-1/sqrt2$ -

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

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

POI , at $x=+-1/sqrt6$ .

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

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