How do I find the upper bound of a function?
Best way is to find it from the graph of the function.
The range of a function can be found in many ways , best way and a concrete way is by plotting its graph and determining.
How you draw the graph depends on you, a generic way is by differentiating the function to get the critical points or the points of maxima or minima . Then you double differentiate the function to see if the critical point is a maxima or minima by putting the critical point in the double differentiated function. If the output is positive then is a minima , if its negative then its a maxima and if its zero then the test fails,we have to go for high derivative tests. The global maxima points give you the upper bound.
Also we can determine the point of inflection for the given curve to determine the convexity or concavity changes of the given function to make the graph more precise .
Thus, having plotted the graph we can tell the upper bound of the function by merely looking at the maximum value of y for all values of x.