If we explore the analysis and graph of functions which are neither polynomial nor rational, we quickly realize the importance of a comprehensive definition for critical points, the various types of critical points, their proper calculation, and their order in the analysis of the function to be graphed. I believe that identifying these values at the earliest part of the analysis is crucial to successful sketching because it eliminates confusion. There is so much relevant information to analyze and accumulate at one time that a set procedure to follow facilitates the process. With this in mind, let us establish the following definitions:
1. Critical Points: The set of all values of x for which the first derivative is either zero or undefined.
2. Extrema or Maxima/Minima: The values of x for which the numerator of the derivative expression is zero or for which the derivative of the polynomial is zero.
3. Vertical Asymptote: The values of x for which the denominator of the given rational expression equals zero.
4. Vertical Tangents: The values of x for which the given expression is defined, but the derivative expression is not defined.
Let us apply these definitions to the following examples beginning with a polynomial: = + - 16 - 16. The critical points for the derivative polynomial = 3+2-16 are x = -8/3 and x = 2. These are extrema only, as there are no vertical asymptotes or vertical tangents for polynomial functions. Rational expressions such as ( 1 ) or ( 2 ) are more interesting because they require more thought. The quick identification of x = -4 in (1) takes that value out of consideration for any other category other than vertical asymptote. However, the derivative of is, and this yields no extrema. Since has already been labeled as a vertical asymptote, we quickly move on to the behavior of the given curve. Here, we include all critical points including all vertical asymptotes and all vertical tangents, in essence, any value of x which causes an interruption of the number line. And, since there are no extrema, the discussion is limited to activity before and after. For, we have a vertical asymptote at (as well as a diagonal asymptote at), and extrema at. Here we equate both and to zero for extrema and leave alone, as it has already been designated a vertical asymptote.
Now consider the non-rational function. Note that is defined for. However, the first derivative yields as a vertical tangent because is undefined. The x-value of -2 becomes a candidate for extrema because it came from the numerator. In further derivatives, we would not be concerned with; whereas had it not been classified, the second derivative would be more difficult to assess. Instead becomes our only point of inflection without concern for.
In our next example,, the derivative is . Here would be a vertical asymptote and there would be no candidates for extrema because. In terms of, remains a vertical asymptote, thereby not confusing the analysis. Thus we need only concentrate on or for our points of inflection.
This clarification on this important topic of critical points is designed to ease the difficulty with curve sketching by allowing the reader to account for every value of x incurred and to use it to its best advantage. I truly believe that an ordered sequence of steps helps the beginner with what can be a complicated procedure. And once the discussion of critical points has taken place, the rest of the analysis is much easier.