The optimization problem is as follows.
Control variable : $x\in[a,b]$
Objective function : $xf(x)$
So the optimization problem is $$\max_{x\in[a,b]}xf(x).$$
What's special about $f$ is that $f$ is a decreasing step function which can be specified by $(x_i,y_i).$ If $x\in(x_i,x_{i+1}],$ then $f(x)=y_{i+1}.$
For example,
$f(x)=1$ if $x\leq x_1$
$f(x)=0.5$ if $x\in(x_1,x_2]$
$f(x)=0.1$ if $x> x_2.$
So, in the example, $y_1=1,y_2=0.5$ and $y_3=0.1$.
In terms of $x_i$ and $y_i$, the optimization problem can be rewritten as $$\max_{i\in\{1,2,\cdots,n\}}x_iy_i.$$
Is there a solution in terms of $y=(y_1,\cdots,y_n)$? What is the name for this specific problem?