TITLE: Is the maximum of the absolute value of the second derivative always smaller than the maximum of the absolute value of the 1st derivative?
QUESTION [0 upvotes]: Assume a function $f(x)$ is differentiable, is the maximum of the absolute value of the second derivative always smaller than the maximum of the absolute value of the 1st derivative?
$$\max\left(\frac{df(x)}{dx}\right) > \max\left(\frac{d^2f(x)}{dx^2}\right)$$ over $x$?

REPLY [6 votes]: No.  A unit argument can show it.  Assuming $f$ is dimensionless and $x$ has units of length, $f'$ has dimensions of inverse length and $f''$ has dimensions of inverse length$^2$, so you can't compare them.  The numeric comparison will depend on the units of measure of $x$.  
For a specific example, take $f(x)=\sin(2x)$ then $\max f'=2, \max f''=4$

REPLY [1 votes]: Simple counterexample: consider the function 
$$f(x)=x^2$$
on the interval $(0,\frac{1}{2})$.
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