SOLUTION: Let c be a real number. What is the maximum value of c such that the graph of the parabola y = 1/5 x^2 has at most one point of intersection with the line y = cx?

      Let' first determine the set of possible values of "c", such that the graph 
      of the parabola y = 1/5 x^2 has at most one point of intersection with the line y = cx?



This parabola and all such lines ALWAYS have at least one intersection (or common) point: 
this point is  (0,0), the origin of the coordinate system.


Next, if  c =/= 0  (i.e. if the straight line has non-zero slope), then certainly
there is another intersection point.



This fact is obvious, if to recall that our parabola (as any other parabola)
raises faster than any non-degenerated linear function that has common point 
with the parabola.- so, another intersection point does exist inevitably.


From this reasoning, you see that there is only one straight line of the form y= cx
which has only one intersection point with the given parabola.


This line is the horizontal line  y = 0  with  c = 0.
The intersection point is the tangent point (0,0), the origin of the coordinate system.


Thus the set of all possible coefficients {c}  consists of one single element c= 0.


Therefore, the ANSWER  to the problem's question is  THIS


    +---------------------------------------------------------------+
    |      the maximum value of c such that the graph of the        |
    |   parabola y = 1/5 x^2 has at most one point of intersection  |
    |           with the line y = cx  is 0 (zero).                  |
    +---------------------------------------------------------------+



Again, the set of all such values "c" consists of one single element 0 (zero),
and the maximum value of all such "c"s is 0, naturally.