121616949-math.108 - 94 Chapter 5 Curve Sketching A B...

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94 Chapter 5 Curve Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................... . . . . . .......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A A B Figure 5.1 Some local maximum points ( A ) and minimum points ( B ). If ( x, f ( x )) is a point where f ( x ) reaches a local maximum or minimum, and if the derivative of f exists at x , then the graph has a tangent line and the tangent line must be horizontal. This is important enough to state as a theorem: THEOREM 5.1 Fermat’s Theorem If f ( x ) has a local extremum at x = a and f is differentiable at a , then f ± ( a ) = 0. Thus, the only points at which a function can have a local maximum or minimum are points at which the derivative is zero, as in the left hand graph in figure 5.1 , or the derivative is undefined, as in the right hand graph. Any value of x for which f ± ( x ) is zero or undefined is called a critical value for f . When looking for local maximum and minimum points, you are likely to make two sorts of mistakes: You may forget that a maximum or minimum can occur where the derivative does not exist, and so forget to check whether the derivative exists everywhere. You might also assume that any place that the derivative
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