Description
Basic topology of the real line, numerical sequences and series, continuity, differentiability, Riemann integration, uniform convergence, power series.
Explanation & Answer
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𝐴𝐷𝑉𝐴𝑁𝐶𝐸𝐷 𝐶𝐴𝐿𝐶𝑈𝐿𝑈𝑆
1.
𝐵, 𝐶
2.
𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑝𝑟𝑜𝑏𝑙𝑒𝑚 𝑖𝑛 𝑡ℎ𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛. 𝑊𝑒 𝑠ℎ𝑜𝑢𝑙𝑑 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 𝑓 ′ (𝑥) ≠ 0 𝑖𝑛 (0,1)𝑡𝑜𝑜.
𝐵𝑒𝑐𝑎𝑢𝑠𝑒 𝑖𝑓 𝑖𝑡 𝑤𝑎𝑠 0 𝑡ℎ𝑒𝑛 𝐶 𝑎𝑛𝑑 𝐷 𝑎𝑟𝑒 𝑏𝑜𝑡ℎ 𝑤𝑟𝑜𝑛𝑔.
So if 𝑓 ′ (𝑥) ≠ 0 𝑎𝑛𝑑 𝑔′ (𝑥) ≠ 0
𝑖𝑡 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 𝑏𝑜𝑡ℎ 𝑓 𝑎𝑛𝑑 𝑔 𝑎𝑟𝑒 𝑆𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑜𝑟 𝑆𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝐷𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠.
𝑓(0) < 𝑓(1) → 𝑓 𝑖𝑠 𝑆𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑖𝑛 (0,1) → 𝑓 ′ (𝑥) > 0
𝑔(0) > 𝑔(1) → 𝑔 𝑖𝑠 𝑆𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝐷𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑖𝑛 (0,1) → 𝑔′ (𝑥) < 0
→ 𝑓 ′ (𝑥)𝑔′ (𝑥) < 0 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 𝑖𝑛 (0,1). 𝑆𝑜 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑓𝑎𝑙𝑠𝑒 𝑜𝑝𝑡𝑖𝑜𝑛.
3.
𝐴: 𝐴(𝑥) = 100𝑥 7 + 2𝑥 3 + 𝑥 2 → 𝐴′ (𝑥) = 700𝑥 6 + 6𝑥 2 + 2𝑥 →
𝐴′′ (𝑥) = 4200𝑥 5 + 12𝑥 + 2 → 𝐴′′ (0) = 2 > 0
𝑆𝑜 𝑖𝑡 ′ 𝑠 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑖𝑛 𝑡ℎ...
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