How do you determine whether the function y=e^x/(5x^2 +1)y=ex5x2+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") :}

Related questions
See all questions in Analyzing Concavity of a Function
Impact of this question
1683 views around the world
You can reuse this answer
Creative Commons License