This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: (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 ) > f prime ( a ) = 0 ,f primeprime ( a ) < 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 ....
View Full Document