Question

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

Answer 1

Refer to explanation

Explanation:

We have that `f(x)=x/(x^2+2)` calculating its second derivative we find that

`d^2 f(x)/(d^2x)=(2x(x^2-6))/(x^2+2)^3`.

So we need to see how the signs change of `2x(x^2-6)` as x goes
from `-oo` to `+oo`

So from `(-oo,-sqrt6]` we have that `f''(x)<0`

from `[-sqrt6,0]` we have that `f''(x)>0`

from `[0,sqrt6]` we have that `f''(x)<0`

from `[sqrt6,+oo)` we have that `f''(x)>0`

In order to determine concavity we use the following theorem

Concavity Theorem:

If the function `f` is twice differentiable at `x=c`, then the graph of f is concave upward at `(c;f(c))` if `f''(c)>0` and concave downward if `f''(c)<0` .