Rog_Sec_14_7_P_2

Rog_Sec_14_7_P_2 - Math 20C Lecture Examples Section 14.7...

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(8/20/08) Math 20C. Lecture Examples. Section 14.7, Part 2. Maxima and minima: The second-derivative test With the Second-Derivative Test for local maxima and minima of functions y = f ( x ) of one variable, we find the points a where the first derivative y prime ( x ) is zero and the tangent line to the graph is horizontal. If f primeprime ( a ) is positive, then the graph of f is concave up at x = a and is above the tangent line for x near a , so the function has a local minimum at that point (Figure 1). If f primeprime ( a ) is negative, then the graph of f is concave down at x = a and below the tangent line for x near x = a and the function has a local maximum there (Figure 2). a y = f ( x ) Tangent line x a y = f ( x ) Tangent line x f prime ( a ) = 0 , f primeprime ( a ) > 0 f prime ( a ) = 0 , f primeprime ( a ) < 0 Local minimum Local maximum FIGURE 1 FIGURE 2 To extend this result to two variables, we need to reinterpret it in terms of Taylor polynomials. Recall that the second-degree Taylor polynomial approximation of y = f ( x ) at x = a is T 2 ( x ) = f ( a ) + f prime ( a )( x - a ) + 1 2 f primeprime ( a )( x - a ) 2 . This polynomial has the same value and first two derivatives as y = f ( x ) at a and, consequently, is the second-degree polynomial that best approximates y = f ( x ) near a . If a is a critical point of y = f ( x ), then f prime ( a ) = 0 and the Taylor polynomial is T 2 ( x ) = f ( a ) + 1 2 f primeprime ( a )( x - a ) 2 .
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