2011 Λύσεις Σχ. β&I

Δ âóâfiìâó μ îè μ âóè ûìûù ôìóˆ

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Δ· ÂӉ¯fiÌÂÓ· ( ∞ – μ ) Î·È ( μ – ∞ ) Â›Ó·È ·Û˘Ì‚›‚·ÛÙ·. ∂Ô̤ӈ˜ ƒ ( ( μ ) ( μ – ∞ ) ) = ƒ ( ∞ – μ ) + ƒ ( μ – ∞ ) = ƒ ( ) – ƒ ( μ ) + ƒ ( μ ) – ƒ ( μ ) = ƒ ( ) + ƒ ( μ ) + 2 ƒ ( μ ) ‰ËÏ·‰‹ 12% . = 10 100 + 6 100 4 100 = 12 100 , = 10 100 + 6 100 2 100 = 14 100 , ƒ ( ) = 10 100 , ƒ ( μ ) = 6 100 Î·È ƒ ( μ ) = 2 100 . = 25 100 + 55 100 15 100 = 65 100 , ƒ ( ) = 25 100 , ƒ ( μ ) = 55 100 , ƒ ( μ ) = 15 100 . = 6 12 + 4 12 1 12 = 9 12 = 3 4 . = 1 2 + 1 3 1 12 = 1 2 + 1 – 2 3 1 12 ∫∂º∞§∞π√ 1: ¶π£∞¡√Δ∏Δ∂™ 10 A B A–B B–A ø
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14. ŒÛÙˆ ÙÔ ÂӉ¯fiÌÂÓÔ Ó· Ì·ı·›ÓÂÈ ·ÁÁÏÈο Î·È μ ÙÔ ÂӉ¯fiÌÂÓÔ Ó· Ì·- ı·›ÓÂÈ Á·ÏÏÈο. Œ¯Ô˘Ì ÕÚ· ƒ ( ( μ ) = 1 – ƒ ( μ ) = 1 – ƒ ( ) – ƒ ( μ ) + ƒ ( μ ) ‰ËÏ·‰‹ 10% . μã √ª∞¢∞™ 1. i) ƒ ( μ ) = ƒ ( ) + ƒ ( μ ) – ƒ ( μ ) = Î + Ï Ì ii) ƒ ( ( μ ) = 1 – ƒ ( μ ) = 1 – Î Ï + Ì iii) ƒ ( ( ∞ – μ) ( μ ) ) = ƒ ( μ ) + ƒ ( μ ) = ƒ ( ) – ƒ ( μ ) + ƒ ( μ ) – ƒ ( μ ) = ƒ ( ) + ƒ ( μ ) – 2 ƒ ( μ ) = Î + Ï 2 Ì . 2. ∞Ó ∞ Î·È μ Ù· ÂӉ¯fiÌÂÓ· Ó· ÌËÓ ¤¯ÂÈ ¤Ó· ÓÔÈÎÔ΢ÚÈfi ÙËÏÂfiÚ·ÛË Î·È μ›ÓÙÂÔ ·ÓÙÈÛÙÔ›¯ˆ˜, ı· Â›Ó·È ∂Ô̤ӈ˜ Ë ˙ËÙÔ‡ÌÂÓË Èı·ÓfiÙËÙ· ı· ›ӷÈ: ƒ ( ( μ ) = 1 – ƒ ( μ ) = 1 – [ ƒ ( ) + ƒ ( μ ) – ƒ ( μ )] ‰ËÏ·‰‹ 55% . 3. Œ¯Ô˘Ì ‰È·‰Ô¯Èο 4 ƒ ( ) = 3 – 3 ƒ ( ) 7 ƒ ( ) = 3, ƒ ( )= 3 7 , ƒ ( ã) = 1 – ƒ ( ) = 4 7 . ƒ ( ) 1 – ƒ ( ) = 3 4 ƒ ( ) ƒ ( ã) = 3 4 = 1 – 15 100 + 40 100 10 100 = 1 – 45 100 = 55 100 , ƒ ( ) = 15 100 Î·È ƒ ( μ ) = 40 100 Î·È ƒ ( μ ) = 10 100 . = 1 – 80 100 30 100 + 20 100 = 10 100 , ƒ ( ) = 80 100 , ƒ ( μ ) = 30 100 Î·È ƒ ( μ ) = 20 100 . 1.2. ŒÓÓÔÈ· Ù˘ Èı·ÓfiÙËÙ·˜ 11
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4. ∞Ó ƒ ( ) = x , ÙfiÙ ƒ ( ã) = 1 – x , fiÔ˘ 0 < x < 1. Œ¯Ô˘Ì 1 – x + x ≥ 4 x (1 – x ) 1 – x + x ≥ 4 x – 4 x 2 4 x 2 – 4 x + 1 ≥ 0 (2 x – 1) 2 ≥ 0 Ô˘ ÈÛ¯‡ÂÈ. 5. ñ Œ¯Ô˘ÌÂ μ ƒ ( μ ) ≤ ƒ ( ) ƒ ( μ ) ≤ 0,6 (1) ñ Œ¯Ô˘Ì ƒ ( μ ) ≤ 1 ƒ ( ) + ƒ ( μ ) – ƒ ( μ ) ≤ 1 0,6 + 0,7 – ƒ ( μ ) ≤ 1 0,6 + 0,7 – 1 ≤ ƒ ( μ ) 0,3 ≤ ƒ ( μ ) (2) ·fi ÙȘ (1) Î·È (2) ÚÔ·ÙÂÈ fiÙÈ: 0,3 ≤ ƒ ( μ ) ≤ 0,6. 6. ƒ ( μ ) – ƒ ( ã) ≤ ƒ ( μ ) ƒ ( μ ) – 1 + ƒ ( ) ≤ ƒ ( μ ) ƒ ( μ ) + ƒ ( ) – ƒ ( μ ) ≤ 1 ƒ ( μ ) ≤ 1 Ô˘ ÈÛ¯‡ÂÈ. 1 x + 1 1 – x ≥ 4 1 ƒ ( ) + 1 ƒ ( ã) ≥ 4 ∫∂º∞§∞π√ 1: ¶π£∞¡√Δ∏Δ∂™ 12
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KEº∞§∞π√ 2 √𠶃∞°ª∞Δπ∫√π ∞ƒπ£ª√π ¨ 2.1. √È Ú¿ÍÂȘ Î·È ÔÈ È‰ÈfiÙËÙ¤˜ ÙÔ˘˜ ∞ã √ª∞¢∞™ 1. Œ¯Ô˘Ì (i) (ii) °È· x = 2010 Î·È ¤¯Ô˘Ì x y = 1 ÔfiÙ ∞ = 1 9 = 1.
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