The Goal:
is to figure out the global extrema of the Minkowski Question Mark function $?(x)$. Here is the graph of:
$$?(x)-x:$$
The $y$ value of the global maximum was found by systematically guessing decimal values:
$y≈0.142522$:
It turns out that:
$$?(.142522)=.0156249999…≈\frac 1{64}\implies ?(y)=\frac1{64}$$
Since $?(x)≈2^{1-\frac1x},y=\frac17$,
where the x coordinate of the maximum is the value such that:
$$?(x)-x\pm\frac17=0$$
Therefore:
$$|?(x)-x|\le\frac17:$$
and the simple bounds for the function itself:
$$x-\frac17≤?(x)≤x+\frac17:$$
but how do we prove the extrema y values? Is this the correct result? Since this function is nondifferentiable, one cannot use a derivative test to find the maximum. Please correct me and give me feedback!
Correction
We have good bounds for the function, but after some iterative zooming and trials from @TheSimpliFire, we got the following result:
$$\min(?(x)-x)=\pm 0.14259056927… \text{at}\ x=0.207075196…,x_2=x_\rm{max}$$
There is the interesting result of almost $\frac1 {64}$:
$$?(0.14259056927)= 0.0156249999999999999999998347561228918792761675351652337679562248… $$
$$?(0.14259056927)-0.14259056927= -0.126965569278000000000000165243877108120723832464834766232043775… $$
$x-?(x)-0.142590569278$:
Second Goal: Main Local Extrema:
Even though the function’s derivatives are $0,\pm\infty$, but Wolfram Alpha plot the derivative of the function:
$$\boxed{\max(?(x)-x)=-\min(?(x)-x)\iff ?’(x)=1}:$$
Therefore the largest local maximum of $?(x)-x$ is about $(0.458211892, 0.0300691122226992386)$ and the minimum has a similar value:
We have an exact form of this maximum as the solution to of $?’(x)-1=0$, but how Wolfram Alpha plots the derivative and this new set of extrema are another problem.
$?’(x):$
We also have this plot for $?(x)$ vs $x\pm 0.0300691122226992386$. Notice the intersection:
