find the concavity of y'=4y-y^3

Question

 I cant find the concavity of this function but I found the increase/decreae and everything else.

Answer 1

If the second derivative is positive, the function is concave upward.   If the second derivative is negative, the function is concave downward.   The second derivative can be found by differentiating the given first order differential equation then substituting for y' .   The result for the second derivative is found to be:

 

  (4y - y³ ) ( 4 - 3 y²)   This has zeros at  -2, - sqrt(4/3) , 0 , sqrt(4/3) , 2

 

A graphing calculator shows that it is positive for y > 2 and negative for y < -2.   It is positive for y between the first two zeros, negative between the the second zero and 0 , positive between 0 and the fourth zero and negative again between the fourth zero and 2.  

 

Of course finding the x values corresponding to these intervals is not so easy.