Suppose you wish to find the maximum value of for . One way to do this without calculus is to massage the expression until one can apply an elementary inequality such as . To apply this particular result we aim to minimise and apply polynomial division.
In the above steps we assume , otherwise is non-positive for . Hence with equality when or .
Another elementary way that applies to quotients of quadratic polynomials is to re-write the expression as a quadratic in :
For fixed this quadratic equation will have 0, 1 or 2 solutions in depending on whether its discriminant is negative, zero, or positive respectively. At any maximum or minimum value of the function, the discriminant will be zero since on one side of the quadratic equation will have a solution (discriminant non-negative) while on the other it will not (discriminant negative). In the image below a maximum is reached at while it is of opposite signs either side of this.
Hence setting the discriminant of the left side of (2) to 0, from which . Hence extrema are at and . We can solve to find that or is the range of the function . This tells us that is a local maximum (illustrated above) and is a local minimum (occurring when ).
One advantage of this method is that unlike elementary calculus, one bypasses the step of finding the corresponding value (i.e. by solving ) before substituting this into the function to find the extremum value for .
Another advantage is that the equation need not be polynomial in . For example below is a plot of . Using the above-mentioned discriminant trick we solve and find the range of the function is when , or . Below is a plot confirming this using WolframAlpha.
The reader is encouraged to try out other examples, for example this method should work for any equation of the form where is a continuous function of . Of course one should also take care in noting when the function is defined before cross-multiplying.
 Find Range of Rational Functions – analyzemath.com