2011 Λύσεις Σχ. β&I

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fiÙ ·˘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·ÊÈ΋ Ù˘ ·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˘ı›· 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›˙˜. 4. °ÓˆÚ›˙Ô˘Ì fiÙÈ i) ŸÙ·Ó · > 0, ÙfiÙÂ Ë ·Ú·‚ÔÏ‹ y = ·x 2 + ‚x + Á Â›Ó·È ·ÓÔȯً ÚÔ˜ Ù· ¿Óˆ, ÂÓÒ fiÙ·Ó · < 0, ÙfiÙÂ Ë ·Ú·‚ÔÏ‹ Â›Ó·È ·ÓÔȯً ÚÔ˜ Ù· οو. ∂Ô̤ӈ˜, 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. ªÂϤÙË Ù˘ Û˘Ó¿ÚÙËÛ˘ f(x) = ·x 2 + ‚x + Á 101
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ii) ΔÔ Á Â›Ó·È Ë ÙÂÙ·Á̤ÓË ÙÔ˘ ÛËÌ›Ԣ ÙÔÌ‹˜ Ù˘ ·Ú·‚ÔÏ‹˜ y = ·x 2 + ‚x + Á Ì ÙÔÓ ¿ÍÔÓ· yãy. ∂Ô̤ӈ˜, 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 Ì Ìˉ¤Ó ¤¯ÂÈ ÙÔ f 4 . iii) ∏ ÙÂÙ·Á̤ÓË Ù˘ ÎÔÚ˘Ê‹˜ ∫ Ù˘ ·Ú·‚ÔÏ‹˜ y = ·x 2 + ‚x + Á ‰›ÓÂÙ·È ·fi ÙÔÓ Ù‡Ô , ÔfiÙ ÈÛ¯‡ÂÈ ‚ = –2· Ø x . ∂Ô̤ӈ˜ ÁÈ· ÙËÓ f 2 Ô˘ ¤¯ÂÈ · < 0 Î·È x > 0, ¤¯Ô˘Ì ‚ > 0, ÁÈ· ÙËÓ f 3 Ô˘ ¤¯ÂÈ · > 0 Î·È x > 0, ¤¯Ô˘Ì ‚ < 0, ÁÈ· ÙËÓ f 4 Ô˘ ¤¯ÂÈ · < 0 Î·È x > 0, ¤¯Ô˘Ì ‚ > 0, ÁÈ· ÙËÓ f 5 Ô˘ ¤¯ÂÈ · < 0 Î·È x > 0, ¤¯Ô˘Ì ‚ > 0, ÁÈ· ÙËÓ f 6 Ô˘ ¤¯ÂÈ · > 0 Î·È x < 0, ¤¯Ô˘Ì ‚ > 0, Î·È ÁÈ· ÙËÓ f 7 Ô˘ ¤¯ÂÈ · < 0 Î·È x < 0, ¤¯Ô˘Ì ‚ < 0. ŒÙÛÈ ¤¯Ô˘Ì 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‹ Ù˘ ‚Ú›- ÛÎÂÙ·È ÛÙÔÓ ¿ÍÔÓ· yãy, ‰ËÏ·‰‹ ·Ó Î·È ÌÔÓÔ ·Ó ∂Ô̤ӈ˜ Ú¤ÂÈ k = – 1.
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