Question

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

Answer 1

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") :}#