1) Match each of the following graphs to the appropriate conditions for Explain. I. y=f ( x ) 0 1 2 II. y=f ( x ) 0 1 2 III. y=f ( x ) 0 1 2 A. f ' ( x )> 0 on ( 0,2 ) , f ' ' ( x ) <0 on ( 0, 1 ) , f ' x> 0 on ( 1,2 ) B. f ' ( x )> 0 on ( 0,1 ) , f ' ( x )< 0 on (1, 2 ) , f ' ' ( x ) >0 on ( 1,2 ) , f ' ' ( x ) <0 on ( 1, 2 ) c. f ' x> 0 on ( 0,1 ) , f ' ( x )< 0 on (1, 2 ) , f ' ' ( x ) >0 on ( 0, 2 ) D. f ' ( x )> 0 on ( 0,1 ) , f ' ( x )< 0 on (1, 2 ) , f ' ' ( x ) <0 on ( 0, 2 ) f' and f '' . E. f ' ( x )< 0 on ( 0,1 ) , f ' ( x )> 0 on (1, 2 ) , f ' ' ( x ) <0 on ( 0, 1 ) , f ' x>0 on ( 1, 2 ) Write the letter that corresponds to the correct graph: (I) __________ (II) __________ (III) __________ 2) Find the critical numbers for the function f ( x )= (A) x=0, (B) x=0, x=1 (C) x=2 (D) 2x x 1 x=1, x=2 (E) None of the above 2) _____________ 3) Find the intervals on the x -axis on which the function f ( x ) is increasing and those on which it is decreasing, where 2 f ( x )= x 1 2 x +1 3) ______________ (A) Decreasing on ( , 0 ) ; Increasing on ( 0, ) . (B) Increasing on (, 0 ) ; Decreasing on ( 0, ) . (C) Increasing on ( ,1 ) ; Increasing on ( 1, ) ; Decreasing on (1, 1 ) . (D) Decreasing on ( ,1 ) ; Increasing on (1, 1 ) ; Decreasing on ( 1, ) . (E) None of the above 4) Find the maximum value of the function f ( x )= x 2+ 9 x3 2 x 4) _____________ (A) 2/3 (B) 1 (C) 7.75 (D) 16 (E) None of the above 5) Engineers want to design an auxiliary pipeline from a new natural gas source; 3 km from the main pipeline, to a power plant located 10 km down the main pipeline, in the least amount of time (see Figure 1). If the oil can flow 10 km/hr along the auxiliary pipeline (OB) and 15 km/hr along the main pipeline (AP), where should the engineers join the auxiliary pipeline to the main pipeline to get the gas to P as soon as possible? Figure 1 Nat ural Gas Sou rce O 3 A x B 1 0 10 -x P Pow er Plant 5) _____________ (A) 0 (B) 0.89 (C) 0.97 (D) 6/ 5 (E) None of the above