see http://www.math.utexas.edu/pipermail/maxima/2013/031439.html
Find roots of f=log(x^3-x*78+90)+1000*x
(%i1) f:log(x^3-x*78+90)+1000*x$
(%i2) allroots(x^3-x*78+90); map(rhs,%)$
(%o2) [x=1.174624074847539,x=8.185668853475939,x=-9.360292928323476]
(%i4) [x1,x2,x3]:sort(%);
(%o4) [-9.360292928323476,1.174624074847539,8.185668853475939]
x^3-x*78+90<0 if x in (-infinity, x1)U (x2, x3). Then
function f is defined for all x in (x1, x2)U(x3, infinity)
(%i5) limit(f,x,x1,plus);
(%o5) -inf
(%i6) limit(f,x,x2,minus);
(%o6) 1140.659863000102
(%i7) limit(f,x,x3,plus);
(%o7) 8155.863524711861
--> wxplot2d([f], [x,-20,20])$
Function f has only one root:
(%i8) find_root(f, x, -1, 0);
(%o8) -0.0045037052827993
Problem for students: find roots of log(x^3-x*78+90)+n*x.
where n is the number of student.
--> with_slider(
a,makelist(i,1,20),
[log(x^3-x*78+90)+a*x],
[x,-20,20],
[legend,false]);
best
Aleksas D