How do you determine whether the function # y=e^x/(5x^2 +1)# is concave up or concave down and its intervals?

1 Answer
Jim H
Oct 20, 2015

You investigate the sign of #y''#

Explanation:

For #y = e^x/(5x^2 +1)#, we get

#y'' = (e^x(25x^4-100x^3+160x^2-20x-9))/(5x^3+1)^3#

The denominator has no real zeros.

Find the real zeros of #25x^4-100x^3+160x^2-20x-9#
and investigate the sign of #y''# on the resulting intervals.

The real zeros of #25x^4-100x^3+160x^2-20x-9# are approximately

#-0.18# and #0.36#. (The other two zeros are imaginary.)

#{: (bb "Interval", bb"Sign of "f'',bb" Concavity"), ((-oo,-0.18)," " +" ", " ""Up"), ((-0.18,0.36), " " - " ", " ""Down"), ((0.36,oo), " " +, " ""Up") :}#

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