
Concavity.
I have a question that asks , ''for what values of x is the function concave?"
Im unsure as to what this is asking, is it just asking if $\displaystyle f''(x) < 0 $ the function is concave. and $\displaystyle f''(x) > 0 $ the function is convex.
??

You'r right with the f''(x)<0, so the question is asking you to figure out which values of x make this true. If the question gives you an actual function that we can work with, find its second derivative and then put it in the abovementioned inequality, and solve for x. You should end up with an inequality for x, like "x > 3" or "x < (2/7)".
For example in the function
$\displaystyle f(x)=4x^{3}+x^{2}$
$\displaystyle f''(x)=24x+1$
so
$\displaystyle 24x+1<0$
$\displaystyle 24x<1$
$\displaystyle x<\frac{1}{24}$
So the original function is concave when x is less than 1/24
Hope this helps