## Transcribed Text

1) Match each of the following graphs to the appropriate conditions for 𝑓′ and 𝑓′′. Explain.
I.
II.
III.
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)
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) __________
0 1 2
0 1 2
𝑦 = 𝑓(𝑥)
𝑦 = 𝑓(𝑥)
0 1 2
𝑦 = 𝑓(𝑥)
2) Find the critical numbers for the function 𝑓(𝑥)=2𝑥√𝑥−1
(A) 𝑥=0,
(B) 𝑥=0,𝑥=1
(C) 𝑥=2
(D) 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 𝑓(𝑥)=𝑥2−1𝑥2+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)=𝑥2+9𝑥−3𝑥2
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
5) _____________
(A) 0
(B) 0.89
(C) 0.97
(D) 6/√5
(E) None of the above
Power Plant
Natural Gas Source
O
A
B
P
x
10
10 – x
3

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