Kaze1

you have to reconsider your atempt to prove that f(x)g(x) is not monotonic.....

I will let you find yourself what I mean

Example 1 f(x)=x^5....and g(x) =1/x then f(x)g(x)=x^4 which is of course monotonic( increasing/decreasing)

Example 2 f(x)=x and g(x)=1/x^2 then f(x)g(x)=1/x which is again monotonic ( decreasing)

example 3 f(x)=x and g(x) =1/x then f(x)g(x)=1 constant function....therefore not monotonic...

therefore the product of two functions are either monotonic or not.....either increasing or decreasing.....

consequently your primay concept is wrong.

meanwhile you cannot use the definition of the monotone increasing or decreasing to prove that a function f(x) is increasing or decreasing because you have to prove this for a large number of points (values) ...

Consult the differential calculus to find a theorem that is associated with the monotone increasing functions and simply says that :if the derivative of a function is positive then the function is a monotone increasing function and if the derivative is negative then the function decreases..... a very well known theorem that is widely used to determine the nature of the stationary points of a function.

MINOAS