SOLUTION: Suppose that f(x) and g(x) are functions such that the range of f is [-5,3], and the range of g is [-2,1]. The range of f(x) * g(x) is [a,b]. What is the largest possible value of

From one side, we have this inequality

    f(x) * g(x) <= |f(x)| * |g(x)|.


It tells us that b <= max(|f(x)| * |g(x)|) <= max|f(x)| * max|g(x)| = 5*2 = 10.    (1)



From the other side,    in some point x,  f(x) = -5, g(x) = -2  ,    f(x)*g(x) = (-5)*(-2) = 10.

Therefore, then b >= 10.    (2)



From these two inequalities, (1) and (2), we conclude that the maximum possible value of b is  (-5)*(-2) = 10.



        It does not mean necessary that max (f(x)*g(x)) is always 10,  
        for any given functions f(x) and g(x), satisfying the condition.



What it means precisely in this problem, are two facts:

    (a)  b can not be greater than 10;  

and

    (b)  for some functions f(x) and g(x) under given conditions, "b" can be 10.