2011 Λύσεις Σχ. β&I

Fiሠûììâùú óùûùôè ûìâúûìù

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§fiÁˆ Û˘ÌÌÂÙÚ›·˜, ·ÓÙ›ÛÙÔȯ· Û˘ÌÂÚ¿ÛÌ·Ù· ¤¯Ô˘ÌÂ Î·È fiÙ·Ó · ≤ 0. 2 2 2 · 2 · 2 (x – ·) 2 + y 2 = 1 d = (x – ·) 2 + (y – 0 2 ) = (x – ·) 2 + y 2 . A™∫∏™∂π™ °π∞ ∂¶∞¡∞§∏æ∏ 110
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11. ∂Âȉ‹ ÙÔ ÙÚ›ÁˆÓÔ ª°§ Â›Ó·È ÔÚıÔ- ÁÒÓÈÔ, ı· ÈÛ¯‡ÂÈ ª° 2 = §° 2 – ª§ 2 = 3 2 – x 2 = 9 – x 2 , ÔfiÙ ı· Â›Ó·È ª¢ = 2ª° = Î·È ÂÂȉ‹ ÙÔ ÙÚ›- ÁˆÓÔ ª∫¢ Â›Ó·È ÔÚıÔÁÒÓÈÔ ı· ÈÛ¯‡ÂÈ ∫¢ 2 = ª∫ 2 + ª¢ 2 6 2 = (3 – x) 2 + 36 = x 2 – 6x + 9 + 4(9 – x 2 ) x 2 + 2x – 3 = 0 x = 1 x = –3 ÕÚ· x = 1, ·ÊÔ‡ x > 0. 12. i) ∞fi ÙÔÓ ÔÚÈÛÌfi Ù˘ ·fiÛÙ·Û˘ ‰˘Ô ÛËÌ›ˆÓ ÙÔ˘ ¿ÍÔÓ· ÚÔ·ÙÂÈ fiÙÈ (ª∞) = |x + 1| Î·È (MB) = |x – 1|. EÔ̤ӈ˜, ¤¯Ô˘Ì f(x) = (ª∞) + (MB) = |x + 1| + |x – 1|. g(x) = | |x + 1| – |x – 1| | . ii) °È· Ó· ·ÏÔÔÈ‹ÛÔ˘Ì ÙÔÓ Ù‡Ô Ù˘ Û˘Ó¿ÚÙËÛ˘ f Î·È g, ‚Ú›ÛÎÔ˘Ì ÙÔ ÚfiÛËÌÔ ÙˆÓ x + 1 Î·È x – 1 ÁÈ· ÙȘ ‰È¿ÊÔÚ˜ ÙÈ̤˜ ÙÔ˘ x Ô˘ Ê·›ÓÔÓÙ·È ÛÙÔÓ ·Ú·Î¿Ùˆ ›Ó·Î·. ŒÙÛÈ ¤¯Ô˘Ì –2x, ·Ó x < –1 2, ·Ó x < –1 f(x) = 2, ·Ó –1 ≤ x <1 Î·È g(x) = –2x, ·Ó –1 ≤ x <1 2x, ·Ó x ≥ 1 2x, ·Ó 0 ≤ x < 1 2, ·Ó x ≥ 1 ÔfiÙ ÔÈ ÁÚ·ÊÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ ÙˆÓ f Î·È g Â›Ó·È ÔÈ ·ÎfiÏÔ˘ı˜: 2 9 – x 2 2 2 9 – x 2 A™∫∏™∂π™ °π∞ ∂¶∞¡∞§∏æ∏ 111 { {
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iii) ∞fi ÙȘ ÚÔËÁÔ‡ÌÂÓ˜ ÁÚ·ÊÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ Û˘ÌÂÚ·›ÓÔ˘Ì fiÙÈ ñ ∏ Û˘Ó¿ÚÙËÛË f Â›Ó·È ÁÓËÛ›ˆ˜ Êı›ÓÔ˘Û· ÛÙÔ (– , –1], ÛÙ·ıÂÚ‹ ÙÔ˘ [–1, 1] Î·È ÁÓË- Û›ˆ˜ ·‡ÍÔ˘Û· ÛÙÔ [1, + ) Î·È ·ÚÔ˘ÛÈ¿˙ÂÈ ÂÏ¿¯ÈÛÙÔ, ›ÛÔ Ì 2, ÁÈ· οı x [–1, 1]. ñ ∏ Û˘Ó¿ÚÙËÛË g Â›Ó·È ÛÙ·ıÂÚ‹ ÛÙÔ (– , –1], ÁÓËÛ›ˆ˜ Êı›ÓÔ˘Û· ÛÙÔ [–1, 0], ÁÓË- Û›ˆ˜ ·‡ÍÔ˘Û· ÛÙÔ [0, 1] Î·È ÛÙ·ıÂÚ‹ ÛÙÔ [1, + ), ·ÚÔ˘ÛÈ¿˙ÂÈ ÂÏ¿¯ÈÛÙÔ, ›ÛÔ Ì 0, ÁÈ· x = 0 Î·È ·ÚÔ˘ÛÈ¿˙ÂÈ Ì¤ÁÈÛÙÔ, ›ÛÔ Ì 2, ÁÈ· οı x (– , –1] [1, + ). 13. i) ñ H f ¤¯ÂÈ ÔÏÈÎfi ̤ÁÈÛÙÔ ÁÈ· x = 0, Ùo f(0) = 2. ñ H g ¤¯ÂÈ ÔÏÈÎfi ̤ÁÈÛÙÔ ÁÈ· x = 1, Ùo g(1) = 2 Î·È ÔÏÈÎfi ÂÏ¿¯ÈÛÙÔ ÁÈ· x = –1, Ùo g(–1) = –2. ñ H h ¤¯ÂÈ ÔÏÈÎfi ̤ÁÈÛÙÔ ÁÈ· x = –1 Î·È x = 1, ÙÔ h(–1) = h(1) = 2 Î·È ÔÏÈÎfi ÂÏ¿¯ÈÛÙÔ ÁÈ· x = 0, ÙÔ h(0) = 0. ii) ñ °È· ÙËÓ f ·ÚΛ Ó· ‰Â›ÍÔ˘Ì fiÙÈ ÁÈ· οı x ÈÛ¯‡ÂÈ f(x) ≤ 2 ≤ 2 1 ≤ x 2 + 1 x 2 ≥ 0 Ô˘ ÈÛ¯‡ÂÈ.
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