There is a minimum at d at d the function is not differentiable Lecture 6 81

There is a minimum at d at d the function is not

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There is a minimum at d (at d the function is not differentiable ) Lecture 6 8.1 Introduction *Extreme points: max and min *Necessary first-order condition *Stationary points 8.2 Simple Tests for Extreme Points *First derivative test for max and min *Max/min for concave/convex functions 8.3 An Economic Example 8.2 Simple tests for extreme points Theorem 1a : First derivative test for maximum . Suppose that a function f is *C1: differentiable in an interval I *C2: c is an interior point of I and f’(c)=0 If '( ) 0 for and '( ) 0 for then is a m axim um point for . f x x c f x x c x c f = 8.2 Example Find the maximum for f(t). 2 ( ) , 0 4 t f t t t = + 2 2 2 1 ( 4) 2 '( ) ( 4) t t t f t t + - + - = + 2 2 2 4 ( 4) t t - + - + - - = + 2 2 2 4 ( 4) t t - = + 2 2 (2 )(2 ) ( 4) t t t - + - = + 8.2 Example Sign diagram!! 2 2 (2 )(2 ) '( ) ( 4) t t f t t - + = + '( ) 0 for 0< 2 and '( ) 0 for 2 f t t f t t > < < > Conclusion: 2 maximizes ( ) c f t = 8.2 Simple tests for extreme points Theorem 1b : First derivative test for minimum . Suppose that a function f is *C1: differentiable in an interval I and *C2: c is an interior point of I and f’(c)=0 If '( ) 0 for and '( ) 0 for then is a minimum point for . f x x c f x x c x c f =

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8.2 Simple tests for extreme points Theorem 2 : max/min for concave/convex functions Suppose that a function f is *C1: differentiable ( 2 times ) in an interval I *C2: c is an interior point of I and f’(c)=0 .

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