Mathematics

3) f (x) = 2 x2 + 4000 x

+ 10

a) F i n d t h e d e r i v at i v e . b ) F i n d t h e cr i t i ca l n u m b e r (s), i f a n y .

4) G i v e n f (x) = 2x ( x2 + 1 )

a) F i n d t h e d e r i v at i v e. b ) F i n d t h e cr i t i ca l n u m b e rs .

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5) L e t f ( x ) = 3×4 + 4×3

a) F i n d t h e d e r i v at i v e. b ) F i n d t h e cr i t i ca l n u m b e rs. c) F i n d t h e a b so l u te e x t re m a f (x) o n [ – 2 , 1]

6) L et f ( x ) = x4 + x3 – 3×2 + 1 a) F i n d a l l t h e h y p e rcr i t i ca l v a l u es. b ) D ete r m i n e t h e i n te r v a l s u p o n w h i c h f (x) i s co n ca v e u p o r co n ca v e d o w n .

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7) L et t h e cost f u n ct i o n b e C (x) = x 2

8 + 4x + 200

a n d t h e u n i t i s so l d at p r i ce o f p = 49 – x d o l l a rs . W h at p r i ce w i l l y i e l d a m a x i m u m p r o f i t ?

8) S u p p ose t h e t ota l cost o f a ce r ta i n p r o d u ct i s C (x) = 0.001 x2 + 5x + 300 . T h e p r o d u ct ca n b e so l d at a p r i ce o f p d o l l a r s, w h e r e p = 8 – 0.003x. I f a l l t h e p r o d u cts a r e so l d , f i n d t h e d a i l y l e v e l o f p r o d u ct i o n t h at w i l l y i e l d a m a x i m u m p r o f i t.

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9) A f i r m i n a co m p et i t i v e m a r k et m u st se l l i ts p r o d u ct f o r $ 200 p e r u n i t. T h e t o ta l C ost p e r m o n t h i s g i v e n b y : C (x) = 80 x + x2 w h e r e x i s t h e n u m b e r o f u n i ts so l d p e r m o n t h . H o w m a n y u n i ts m u st b e so l d t o m a x i m i z e p r o f i t? W h at i s t h e m a x i m u m p r o f i t ?

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10) A n o p e n b o x i s t o b e m a d e f r o m a 16 i n c h es l o n g a n d 10 i n c h es w i d e r ecta n g u l a r p i ece o f m eta l b y c u tt i n g e q u a l s q u a r es f r o m t h e co r n e rs a n d t u r n i n g u p t h e s i d es. F i n d t h e v o l u m e o f t h e l a r g est b o x t h at ca n b e m a d e i n t h i s m a n n e r