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# È ùë ûóúùëûë gx 2x 2 8x 9 âóè 2 0

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ÂÓÒ ÙÔÓ ¿ÍÔÓ· yãy ÛÙÔ ÛËÌÂ›Ô °(0, 1). ‚) °È· ÙË Û˘Ó¿ÚÙËÛË g(x) = –2x 2 + 8x – 9 Â›Ó·È · = –2 < 0, ÔfiÙÂ ·˘Ù‹ ¶·ÚÔ˘ÛÈ¿˙ÂÈ Ì¤ÁÈÛÙÔ ÁÈ· ∂›Ó·È ÁÓËÛ›ˆ˜ ·‡ÍÔ˘Û· ÛÙÔ (– , 2] Î·È ÁÓËÛ›ˆ˜ Êı›ÓÔ˘Û· ÛÙÔ [2, + ). ∞ÎfiÌË Ë ÁÚ·ÊÈÎ‹ ÙË˜ ·Ú¿ÛÙ·ÛË Â›Ó·È ·Ú·‚ÔÏ‹ Î·È ¤¯ÂÈ ÎÔÚ˘Ê‹ ÙÔ ÛËÌÂ›Ô ∫(2, –1) Î·È ¿ÍÔÓ· Û˘ÌÌÂÙÚ›·˜ ÙËÓ Â˘ıÂ›· x = 2, Ù¤ÌÓÂÈ ÙÔÓ ¿ÍÔÓ· yãy ÛÙÔ ÛËÌÂ›Ô ∞(0, –9) ÂÓÒ, ‰ÂÓ Ù¤ÌÓÂÈ ÙÔÓ ¿ÍÔÓ· xãx, ÁÈ·Ù› ÙÔ ÙÚÈÒÓ˘ÌÔ ‰ÂÓ ¤¯ÂÈ Ú›˙Â˜. 4. °ÓˆÚ›˙Ô˘ÌÂ fiÙÈ i) ŸÙ·Ó · > 0, ÙfiÙÂ Ë ·Ú·‚ÔÏ‹ y = ·x 2 + ‚x + Á Â›Ó·È ·ÓÔÈ¯Ù‹ ÚÔ˜ Ù· ¿Óˆ, ÂÓÒ fiÙ·Ó · < 0, ÙfiÙÂ Ë ·Ú·‚ÔÏ‹ Â›Ó·È ·ÓÔÈ¯Ù‹ ÚÔ˜ Ù· Î¿Ùˆ. ∂ÔÌ¤Óˆ˜, ıÂÙÈÎfi · ¤¯Ô˘Ó Ù· ÙÚÈÒÓ˘Ì· f 1 , f 3 Î·È f 6 , ÂÓÒ ·ÚÓËÙÈÎfi · ¤¯Ô˘Ó Ù· ÙÚÈÒÓ˘Ì· f 2 , f 4 , f 5 Î·È f 7 . x = – = 2, ÙÔ g(2) = –1 A 2 + 2 2 , 0 Î·È μ –2 + 2 2 , 0 x = – = –1, ÙÔ f(–1) = –1. x = – = –5 6 , ÙÔ g –5 6 = –3 –5 6 2 – 5 –5 6 + 2 = 49 12 . 7.3. ªÂÏ¤ÙË ÙË˜ Û˘Ó¿ÚÙËÛË˜ f(x) = ·x 2 + ‚x + Á 101

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ii) ΔÔ Á Â›Ó·È Ë ÙÂÙ·ÁÌ¤ÓË ÙÔ˘ ÛËÌÂ›Ô˘ ÙÔÌ‹˜ ÙË˜ ·Ú·‚ÔÏ‹˜ y = ·x 2 + ‚x + Á ÌÂ ÙÔÓ ¿ÍÔÓ· yãy. ∂ÔÌ¤Óˆ˜, ıÂÙÈÎfi Á ¤¯Ô˘Ó Ù· ÙÚÈÒÓ˘Ì· f 1 Î·È f 5 , ·ÚÓË- ÙÈÎfi Á ¤¯Ô˘Ó Ù· ÙÚÈÒÓ˘Ì· f 2 , f 3 , f 6 Î·È f 7 , ÂÓÒ Á ›ÛÔÓ ÌÂ ÌË‰¤Ó ¤¯ÂÈ ÙÔ f 4 . iii) ∏ ÙÂÙ·ÁÌ¤ÓË ÙË˜ ÎÔÚ˘Ê‹˜ ∫ ÙË˜ ·Ú·‚ÔÏ‹˜ y = ·x 2 + ‚x + Á ‰›ÓÂÙ·È ·fi ÙÔÓ Ù‡Ô , ÔfiÙÂ ÈÛ¯‡ÂÈ ‚ = –2· Ø x . ∂ÔÌ¤Óˆ˜ ÁÈ· ÙËÓ f 2 Ô˘ ¤¯ÂÈ · < 0 Î·È x > 0, ¤¯Ô˘ÌÂ ‚ > 0, ÁÈ· ÙËÓ f 3 Ô˘ ¤¯ÂÈ · > 0 Î·È x > 0, ¤¯Ô˘ÌÂ ‚ < 0, ÁÈ· ÙËÓ f 4 Ô˘ ¤¯ÂÈ · < 0 Î·È x > 0, ¤¯Ô˘ÌÂ ‚ > 0, ÁÈ· ÙËÓ f 5 Ô˘ ¤¯ÂÈ · < 0 Î·È x > 0, ¤¯Ô˘ÌÂ ‚ > 0, ÁÈ· ÙËÓ f 6 Ô˘ ¤¯ÂÈ · > 0 Î·È x < 0, ¤¯Ô˘ÌÂ ‚ > 0, Î·È ÁÈ· ÙËÓ f 7 Ô˘ ¤¯ÂÈ · < 0 Î·È x < 0, ¤¯Ô˘ÌÂ ‚ < 0. ŒÙÛÈ ¤¯Ô˘ÌÂ ÙÔÓ ·Ú·Î¿Ùˆ ›Ó·Î· √ª∞¢∞™ 1. i) ∏ ·Ú·‚ÔÏ‹ ÂÊ¿ÙÂÙ·È ÙÔ˘ xãx ÌfiÓÔ ·Ó Â›Ó·È ¢ = 0. ¢ËÏ·‰‹ (k + 1) 2 – 4k = 0 k 2 + 2k + 1 – 4k = 0 k 2 –2k + 1 = 0 (k – 1) 2 = 0 k = 1. ii) H ·Ú·‚ÔÏ‹ ¤¯ÂÈ ÙÔÓ yãy ¿ÍÔÓ· Û˘ÌÌÂÙÚ›·˜ ÌfiÓÔ ·Ó Ë ÎÔÚ˘Ê‹ ÙË˜ ‚Ú›- ÛÎÂÙ·È ÛÙÔÓ ¿ÍÔÓ· yãy, ‰ËÏ·‰‹ ·Ó Î·È ÌÔÓÔ ·Ó ∂ÔÌ¤Óˆ˜ Ú¤ÂÈ k = – 1.
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