MA1102 Calculus Tutorial 6

Topics Covered:

The three important theorems. Extreme Value Theorem,

Mean Value Theorem and Rolle's Theorem. Relative maxima and relative minima. Stationary points and critical points. Consequence of differentiability. Absolute maxima and absolute minima.

Textbook: Chapter 6

These three theorems are very important tools, in particular for the analysis of the global behaviour of a function. The extreme value theorem guarantees the existence of absolute maximum and absolute minimum of a continuous function on a closed and bounded interval. This is a global information. Mean value theorem links the behaviour of a function f with its derivative. Rolle's theorem is just a special case of the Mean value theorem but is used equally often. Go through the proofs of some of the results in the lectures now and again, you will find that invariably Mean value Theorem is used. A mechanical procedure for finding the absolute extrema of a continuous function is required for some of the questions. In most cases this gives an easy procedure but you may, in general modify and fine tune the procedure for the number of critical points may not always be finite.