Finding the concavity of a function without having to plot I am given a function $f(x)$.
I determined that $f(x)'' = 0$ precisely when $x$ is $4$ or $-3$.
I am asked to find the interval for which the function is concave down.
How can I do it by knowing the values $x = 4$ and $x = -3$ and without having to plot the function?
 A: For simplicity let us assume that the second derivative is everywhere defined and continuous. You have found that $f''(x)=0$ at $x=4$ and at $x=-3$. That is not enough to determine concavity, it only locates the points where concavity might change.
In most simple cases, concavity will change at these points, but it need not. To determine concavity, you need to examine the signs of $f''(x)$ is the intervals $(-\infty,-3)$, $(-3,4)$, and $(4,\infty)$. (Under our conditions, the sign of $f''(x)$ can only change at $-3$ and $4$.)
If for example $f''(x)=(x+3)(x-4)$, then $f''(x)$ is positive in $(-\infty,-3)$, negative in $(-3,4)$, and positive in $(4,\infty)$, so we get concave up, then down, then up.
However, if $f''(x)=(x+3)^2(x-4)$, the story changes: we get down, down, up. And in the case $f''(x)=(x+3)^2(x-4)^2$, we get up, up, up. 
A: If $f''$ is continuous, then you only need to know it at three more points: one less than $-3$, one between $-3$ and $4$, and one greater than $4$. Do you see why you need these? If $f''$ is not continuous, then you may need to know much more.
