Curvature, Maxima, and Minima

Putting in the definition (6.12) of the first derivative, this can also be written as 

This formula is convenient for numerical evaluation of second derivatives. For analytical purposes, we can simply apply all the derivative techniques of Sections 6.4 and 6.5 to the function fix). Higher derivatives can be defined 

These will be used in the following chapter to obtain power series representations for functions. 

Recall that the first derivative fix) is a measure of the instantaneous slope of the function fix) at x. When fix) 0, the function is increasing 

The second derivative f x) is analogously a measure of the increase or decrease in the slope f x). When f x) 0, the slope increases with x and the function has an upward curvature. It is concave upward and would hold water if it were a cup. Conversely, when f (x) 0, the function must have a downward curvature. It is concave downward and water would spill out. A point where f x) = 0, where the curvature is zero, is known as an inflection point. Most often, for a continuous function, an inflection point represents a point of transition between positive and negative curvatures. 

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