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# Fiáˆ ûììâùú óùûùôè ûìâúûìù

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§fiÁˆ Û˘ÌÌÂÙÚ›·˜, ·ÓÙ›ÛÙÔÈ¯· 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. ∂ÂÈ‰‹ 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fiÙÂ ı· Â›Ó·È ª¢ = 2ª° = Î·È ÂÂÈ‰‹ 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 ÙË˜ ·fiÛÙ·ÛË˜ ‰˘Ô ÛËÌÂ›ˆÓ ÙÔ˘ ¿ÍÔÓ· ÚÔÎ‡ÙÂÈ fiÙÈ (ª∞) = |x + 1| Î·È (MB) = |x – 1|. EÔÌ¤Óˆ˜, ¤¯Ô˘ÌÂ f(x) = (ª∞) + (MB) = |x + 1| + |x – 1|. g(x) = | |x + 1| – |x – 1| | . ii) °È· Ó· ·ÏÔÔÈ‹ÛÔ˘ÌÂ ÙÔÓ Ù‡Ô ÙË˜ Û˘Ó¿ÚÙËÛË˜ f Î·È g, ‚Ú›ÛÎÔ˘ÌÂ ÙÔ ÚfiÛËÌÔ ÙˆÓ x + 1 Î·È x – 1 ÁÈ· ÙÈ˜ ‰È¿ÊÔÚÂ˜ ÙÈÌ¤˜ ÙÔ˘ x Ô˘ Ê·›ÓÔÓÙ·È ÛÙÔÓ ·Ú·Î¿Ùˆ ›Ó·Î·. ŒÙÛÈ ¤¯Ô˘ÌÂ –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fiÙÂ ÔÈ ÁÚ·ÊÈÎ¤˜ ·Ú·ÛÙ¿ÛÂÈ˜ ÙˆÓ f Î·È g Â›Ó·È ÔÈ ·ÎfiÏÔ˘ıÂ˜: 2 9 – x 2 2 2 9 – x 2 A™∫∏™∂π™ °π∞ ∂¶∞¡∞§∏æ∏ 111 { {
iii) ∞fi ÙÈ˜ ÚÔËÁÔ‡ÌÂÓÂ˜ ÁÚ·ÊÈÎ¤˜ ·Ú·ÛÙ¿ÛÂÈ˜ Û˘ÌÂÚ·›ÓÔ˘ÌÂ 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 ÌÂ 2, ÁÈ· Î¿ıÂ 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 ÌÂ 0, ÁÈ· x = 0 Î·È ·ÚÔ˘ÛÈ¿˙ÂÈ Ì¤ÁÈÛÙÔ, ›ÛÔ ÌÂ 2, ÁÈ· Î¿ıÂ x (– , –1] [1, + ). 13. i) ñ H f ¤¯ÂÈ ÔÏÈÎfi Ì¤ÁÈÛÙÔ ÁÈ· x = 0, Ùo f(0) = 2. ñ H g ¤¯ÂÈ ÔÏÈÎfi Ì¤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 Ì¤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 ·ÚÎÂ› Ó· ‰Â›ÍÔ˘ÌÂ fiÙÈ ÁÈ· Î¿ıÂ x ÈÛ¯‡ÂÈ f(x) ≤ 2 ≤ 2 1 ≤ x 2 + 1 x 2 ≥ 0 Ô˘ ÈÛ¯‡ÂÈ.

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