How do you determine whether the function #f^('')(x) = 36x^2-12# is concave up or concave down and its intervals?
You can use the second derivative test.
You can determine the intervals on which your function is concave up and concave down by using the second derivative test.
ALl you have to do is examine the behavior of the second derivative around inflexion points, which are points for which the second derivative is equal to zero.
In your case, you have
Take the square root of both sides to get
You have two inflexion points,
SInce you're dealing with a square number, the sign of
This means that
This time, the expression
Once again, the square of
So, your function will be concave up on
The graph of