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Unformatted text preview: range of f. ◦ 7. The point x=a determines an inflection point for function f if f is continuous at x=a , and the second derivative f'' is negative () for x<a and positive (+) for x>a , or if f'' is positive (+) for x<a and negative () for x>a. ◦ 8. THE SECOND DERIVATIVE TEST FOR EXTREMA (This can be used in place of statements 5. and 6.) : Assume that y=f(x) is a twicedifferentiable function with f'(c)=0 . ◦ a.) If f''(c)<0 then f has a relative maximum value at x=c . ◦ b.) If f''(c)>0 then f has a relative minimum value at x=c . Example: f(x) = x 3 3x 2 = 3x (x  2) = 0 x=2, x=0 f''(x) = 6x  6 = 6 (x  1) = 0 FROM f' : f is ( ) for x<0 and x>2 ; f is ( ) for 0<x<2 ; f has a relative maximum at x=0 , y=0 ; f has a relative minimum at x=2 , y=4 . FROM f'' : f is ( ∪ ) for x>1 ; f is ( ∩ ) for x<1 ; f has an inflection point at x=1 , y=2 ....
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 Fall '08
 JUNGHENN
 Calculus, Derivative, Mathematical analysis, Convex function, point x=a

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