Help understanding question regarding 3rd derivative and "smallest uniform bound"? I'm a big user of Stack Overflow, however, a first time user here. I'm working on a problem for a math class that's pretty easy (I'm sure), I just don't understand the question really. Here it is below:
5. Consider g(x) = x^-2

   a) Find g'''(x) (3rd derivative... I can do this)

   b) For -4 ≤ x ≤ -2 , find the smallest uniform bound M such that 

        |g'''(x)| ≤ M for all x such that -4 ≤ x ≤ -2

It's the second part that I don't understand. I don't really know what they mean by "smallest uniform bound". Could anyone please help me understand this problem? Thanks. 
 A: Uniform refers to the fact that the bound $M$ doesn't depend on $x$ (duh! otherwise, you could just one $g'''(x)$). Since the sentence is immediately followed by a precise definition of what's required of $M$, i.e. that $$
  |g'''(x)| \leq M \text{ for all $-4 \leq x \leq -2$,}
$$
the phrasing is a bit redundant - the definitions already tells you that $M$ needs to work for all $x$ in the interval $[-4,-2]$. It's probably written that way to introduce you to the phrase uniform bound, and to emphasize that fact that it's uniform.
To find such a bound, look at $|g'''(x)|$, and see how large it's value can get in the worst case within $[-4,-2]$. One way to do that is to find zeros of the derivative of $g'''(x)$, i.e. of $g''''(x)$. But be carefull - not every such $x$ is a maximum or minimum of $g'''(x)$, and not every maximum or minimum can be found that way. You'll need to check the borders of the interval, plus any $x$ where $g'''(x)$ isn't differentiable separately.
A: Find the maximum value of $|g'''(x)| = 24 |x|^{-5}$ on the interval $[-4,-2]$.
