Section4-2 - Section 4.2 Maximum and Minimum Values Goals...

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1 Section 4.2 Maximum and Minimum Values • Goals – Solve problems requiring the minimum or maximum value of a quantity – Study absolute vs. local maxima/minima of a function – Introduce the Extreme Value Theorem and Fermat’s Theorem , as well as critical points . Optimization Problems These are problems in which we are required to find the optimal (best) way of doing something. Some examples: – What is the shape of a can that minimizes manufacturing costs? – At what angle should blood vessels branch so as to minimize the energy expended by the heart in pumping blood? Absolute Maxima/Minima A function f has an absolute (or global ) maximum at c if f ( c ) f ( x ) for all x in the domain D of f . – The number f ( c ) is called the maximum value of f on D . – Similarly for absolute minimum . – The maximum and minimum values of f are called the extreme values of f . Absolute Maxima/Minima (cont’d) The figure on the next slide shows the graph of a function f with… – absolute maximum at d and – absolute minimum at a . Note that… – ( d , f ( d )) is the highest point on the graph and – ( a , f ( a )) is the lowest point. Absolute Maxima/Minima (cont’d) Local Maxima/Minima In Fig. 1… – if we consider only values of x near b , – then f ( b ) is the largest of those value s of f ( x ) , – and is called a local maximum value of f : A function f has a local (or relative ) maximum at c if f ( c ) f ( x ) when x is near c .
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