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# Ôfiùâ ìôˆ fi ùô ùúáˆóô μ úóôìâ

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, ÔfiÙÂ √ÌÔ›ˆ˜ ·fi ÙÔ ÙÚ›ÁˆÓÔ §¢μ ·›ÚÓÔ˘ÌÂ ΔÔ ¿ıÚÔÈÛÌ· ÙˆÓ ÂÌ‚·‰ÒÓ ÙˆÓ ‰‡Ô ÙÚÈÁÒÓˆÓ Â›Ó·È ÙfiÙÂ ∂ = ∂ 1 + ∂ 2 = ÕÚ· , ÌÂ 0 ≤ x ≤ 6. (1) ∞fi ÙËÓ (1) Û˘ÌÂÚ·›ÓÔ˘ÌÂ fiÙÈ ÙÔ ÂÌ‚·‰fiÓ ∂ Â›Ó·È ÂÏ¿¯ÈÛÙÔ ÁÈ· ÙËÓ ÙÈÌ‹ ÙÔ˘ x, ÁÈ· ÙËÓ ÔÔ›· Ë Û˘Ó¿ÚÙËÛË f(x) = x 2 – 6x + 18 ·ÚÔ˘ÛÈ¿˙ÂÈ ÂÏ¿¯È- ÛÙÔ. ∂ÂÈ‰‹ · = 1 > 0, Ë Û˘Ó¿ÚÙËÛË ·ÚÔ˘ÛÈ¿˙ÂÈ ÂÏ¿¯ÈÛÙÔ ÁÈ· ∂ÔÌ¤Óˆ˜ ÙÔ ÂÌ‚·‰fiÓ Á›ÓÂÙ·È ÂÏ¿¯ÈÛÙÔ fiÙ·Ó ÙÔ ª Â›Ó·È ÙÔ Ì¤ÛÔ ÙÔ˘ ∞μ. 5. ∞fi ÙÔ Û¯‹Ì· ‚Ï¤Ô˘ÌÂ fiÙÈ ÁÈ· ÙÈ˜ ‰È·ÛÙ¿ÛÂÈ˜ x Î·È y ÈÛ¯‡ÂÈ 2x + 2x + 3y = 240 4x + 3y = 240 y = (1) TÔ ÂÌ‚·‰fiÓ ÙˆÓ ‰‡Ô ¯ÒÚˆÓ Â›Ó·È ∂ = 2xy = 2x (2) °È· ÙË Û˘Ó¿ÚÙËÛË Â›Ó·È ÔfiÙÂ ·˘Ù‹ ·ÚÔ˘ÛÈ¿˙ÂÈ Ì¤ÁÈÛÙÔ ÁÈ· ΔfiÙÂ ·fi ÙËÓ (1) ·›ÚÓÔ˘ÌÂ ÕÚ·, ÔÈ ‰È·ÛÙ¿ÛÂÈ˜ Ô˘ ‰›ÓÔ˘Ó ÙÔ Ì¤ÁÈÛÙÔ ÂÌ‚·‰fiÓ Â›Ó·È x = 30m Î·È y = 40m. y = 240 – 4 30 3 = 40. x = –‚ = –160 –16 3 = 30. · = – 8 3 < 0, ∂(x) = – 8 3 x 2 + 160x 240 – 4x 3 = – 8 3 x 2 + 160x. 240 – 4x 3 . x = –‚ = 6 2 = 3. E = 3 2 (x 2 – 6x + 18). = 3 4 x 2 + 3 4 (6 – x) 2 = 1 2 x x 3 2 + 1 2 (6 –x) (6 –x) 3 2 1 2 (∞ª)(∫°) + 1 2 (ªμ)(§¢) ˘ 2 = (6 – x) 3 2 . ˘ 1 = x 3 2 . ˘ 1 2 = x 2 x 2 2 = x 2 x 2 4 = 3x 2 4 ∫∂º∞§∞π√ 7: ª∂§∂Δ∏ μ∞™π∫ø¡ ™À¡∞ƒΔ∏™∂ø¡ 104

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∞™∫∏™∂π™ °π∞ ∂¶∞¡∞§∏æ∏ 1. i) Œ¯Ô˘ÌÂ ii) Œ¯Ô˘ÌÂ · 2 + ‚ 2 + Á 2 ≥ ·‚ + ‚Á + Á· · 2 + ‚ 2 + Á 2 – ·‚ – ‚Á – Á· ≥ 0 Ô˘ ÈÛ¯‡ÂÈ. ΔÔ “=” ÈÛ¯‡ÂÈ ·Ó Î·È ÌfiÓÔ ·Ó · – ‚ = 0 Î·È ‚ – Á = 0 Î·È Á – · = 0 · = ‚ = Á. 2. i) Œ¯Ô˘ÌÂ (Î‚) 2 + (ÎÁ) 2 = Î 2 2 + Î 2 Á 2 = Î 2 (‚ 2 + Á 2 ) = Î 2 · 2 = (Î·) 2 . ii) Œ¯Ô˘ÌÂ (Ì 2 – Ó 2 ) 2 + (2ÌÓ) 2 = Ì 4 – 2Ì 2 Ó 2 + Ó 4 + 4Ì 2 Ó 2 = Ì 4 + 2Ì 2 Ó 2 + Ó 4 = (Ì 2 + Ó 2 ) 2 . 3 4 5 8 6 10 5 12 13 21 20 29 16 30 34 15 8 17 1 2 (· – ‚) 2 + (‚ – Á) 2 + (Á – ·) 2 ≥ 0 = · 2 + ‚ 2 + Á 2 – ·‚ – ‚Á – Á·. = 1 2 2 · 2 + ‚ 2 + Á 2 – ·‚ – ‚Á – Á· = 1 2 2 + 2‚ 2 + 2Á 2 – 2·‚ – 2‚Á – 2Á· = 1 2 · 2 – 2·‚ + ‚ 2 + ‚ 2 – 2‚Á + Á 2 + Á 2 – 2Á· + · 2 1 2 (· – ‚) 2 + (‚ – Á) 2 + (Á – ·) 2
3. ∞) Œ¯Ô˘ÌÂ 4·‚ ≤ · 2 + ‚ 2 + 2·‚ 0 ≤ · 2 + ‚ 2 + 2·‚ – 4·‚ 0 ≤ · 2 + ‚ 2 – 2·‚ 0 ≤ (· – ‚) 2 , Ô˘ ÈÛ¯‡ÂÈ. ΔÔ “=” ÈÛ¯‡ÂÈ fiÙ·Ó · = ‚. ∞fi ÙËÓ ·ÓÈÛfiÙËÙ· ·˘Ù‹ ÚÔÎ‡ÙÂÈ fiÙÈ ÙÔ ÂÌ‚·‰fiÓ ÂÓfi˜ ÔÚıÔÁˆÓ›Ô˘ ÌÂ ‰È·ÛÙ¿ÛÂÈ˜ · Î·È ‚ ‰ÂÓ ˘ÂÚ‚·›ÓÂÈ ÙÔ ÂÌ‚·‰fiÓ ÙÔ˘ ÙÂÙÚ·ÁÒÓÔ˘ ÌÂ ÏÂ˘Ú¿ ÙÔ ËÌÈ¿ıÚÔÈÛÌ·

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• Winter '09
• Nikos

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