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

1 Answer
Jan 18, 2017

Answer:

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

Explanation:

End behavior L

#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]}