2 3 6 5 x 03 f x x x 6 6 6 1 f x x x 1 2 0 5 and 3 14 f f f 85 Economic example

2 3 6 5 x 03 f x x x 6 6 6 1 f x x x 1 2 0 5 and 3 14

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2 ( ) 3 6 5, x [0,3] f x x x = - + - + '( ) 6 6 6( 1) f x x x = - = - = - (1) 2, (0) 5 and (3) 14 f f f = = = = 8.5 Economic example I A firm obtains a fixed price P =121 per unit. Cost function The firm can produce at most Q =110 units. Which Q maximizes profit? The interval: [0,110] is closed bounded . π is differentiable, so continuous . so π must have an maximum point in [0,110]. 3 2 ( ) 0.02 3 175 500 C Q Q Q Q = - + + - + ( ) ( ) ( ) ( ) Q R Q C Q PQ C Q π = - = - = 3 2 121 (0.02 3 175 500) Q Q Q Q = - - + + = -
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2 8.5 Economic example I (i) Find all the stationary points of f in ( a , b ) (ii) Evaluate f at the endpoints a and b of the interval and also all stationary points the stationary points: Q =10, Q =90 and the endpoints Q =0, Q =110. (iii) Conclusion : maximum profit: producing Q*=90 units. 3 2 ( ) 121 (0.02 3 175 500) Q Q Q Q Q π = - - + + = - - + + + + 2 '( ) 121 (0.06 6 175) 0 Q Q Q π = - - + = - 10 and 90 Q Q = = (0) 500, (10) 760, (90) 4360, (110) 3240 π π π π π π = - = - = - = - = - = - = = = Neighbourhood Definition: A neighbourhood of a real number a is an open interval which contains a. Remark: a belongs to the neighbourhood! Examples: 1, ( 3,0), ( 1.1, 0.9), ( 1.0001, 0.999) a = - - - - - - = - - - - - - - - - - - - 4, (3,7), (3.5,4.1) a = 8.6 Local extreme points A function has a local maximum (minimum) at c if there exists a neighbourhood (a,b) of c such that ( ) ( ) ( ( ) ( )) for all in ( , ). f x f c f x f c x a b 8.6 Theorem: first-derivative test for local extreme points How can we determine whether a given stationary point is a local maximum, a local minimum? Let f be a differentiable function in an interval I and let c be an interior point of I . Suppose c is a stationary point for f ( x ). (a) If f ’( x ) 0 on some interval ( a , c ) and f ’( x ) 0 on some interval ( c , b ), then x = c is a local maximum point for f . 8.6 Theorem: first-derivative test for local extreme points (b) If f ’( x ) 0 on some interval ( a , c ) and f ’( x ) 0 on some interval ( c , b ), then x = c is a local minimum point for
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