An example of the realroots function is

block([numer:true],realroots(fff:x^4+3*x^3+1,10^-7)) returns
Result

The numer flag says give a numerical (floating point) answer. Now
the maximum must occur either at a critical point or at an end point
of the interval, so we can tell write a function to find the maximum
on each little subinterval. Note the graph of the above function fff is:
plot2d(fff,[x,-3,3],[y,-10,5]) returns
result