lec11 - Lecture 11 18.01 Fall 2006 Lecture 11 Max/Min...

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Lecture 11 18.01 Fall 2006 Lecture 11: Max/Min Problems Example 1. y = ln x (same function as in last lecture) x x 0 =e 1/e Figure 1: Graph of y = ln x . x 1 What is the maximum value? Answer: y = . e Where (or at what point) is the maximum achieved? Answer: x = e . (See Fig. 1).) Beware: Some people will ask “What is the maximum?”. The answer is not e . You will get so used to finding the critical point x = e , the main calculus step, that you will forget to find the maximum 1 1 value y = . Both the critical point x = e and critical value y = are important. Together, they e e 1 form the point of the graph ( e, ) where it turns around. e Example 2. Find the max and the min of the function in Fig. 2 Answer: If you’ve already graphed the function, it’s obvious where the maximum and minimum values are. The point is to find the maximum and minimum without sketching the whole graph. Idea : Look for the max and min among the critical points and endpoints.You can see from Fig. 2 that we only need to compare the heights or y -values corresponding to
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