1) Match each of the following graphs to the appropriate conditions...

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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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