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# Ôìóˆ ı ôìâ ñ x 2 x 0 x 1 ñ x 2 0 x 1 ii

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∂ÔÌ¤Óˆ˜, ı· ¤¯Ô˘ÌÂ ñ ≤ x 2 x < 0 x ≥ 1. ñ > x 2 0 < x < 1. ii) Œ¯Ô˘ÌÂ ñ ≤ x 2 – x 2 ≤ 0 ≤ 0 ≥ 0 x(x 3 – 1) ≥ 0 Î·È x ≠ 0 x(x – 1)(x 2 + x + 1) ≥ 0 Î·È x ≠ 0 x(x – 1) ≥ 0 Î·È x ≠ 0 x < 0 x ≥ 1. x 3 – 1 x 1 – x 3 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 – x x 1 x 1 x 1 – x x 1 x 1 x 7.2. ªÂÏ¤ÙË ÙË˜ Û˘Ó¿ÚÙËÛË˜ f(x) = · x 99 {

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∂ÔÌ¤Óˆ˜ ñ > x 2 0 < x < 1. 6. ™Â ¤Ó· Û‡ÛÙËÌ· Û˘ÓÙÂÙ·ÁÌ¤ÓˆÓ ·›Ú- ÓÔ˘ÌÂ ∞μ = √∞ = x > 0 Î·È ∞° = √° = y > 0. ΔfiÙÂ ÙÔ ÂÌ‚·‰fi ∂ ÙÔ˘ ÙÚÈÁÒ- ÓÔ˘ Â›Ó·È ∂ = , ÔfiÙÂ ¤¯Ô˘ÌÂ = 2 xy = 4 , (1). H ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙË˜ (1) Â›Ó·È ˘ÂÚ‚ÔÏ‹ ÌÂ ÂÍ›ÛˆÛË Î·È Ê·›- ÓÂÙ·È ÛÙÔ Û¯‹Ì·. ¨ 7.3. ªÂÏ¤ÙË ÙË˜ Û˘Ó¿ÚÙËÛË˜ f(x) = ·x 2 + ‚x + Á ∞ã √ª∞¢∞™ 1. i) Œ¯Ô˘ÌÂ f(x) = 2(x 2 – 2x) + 5 = 2(x 2 – 2 Ø x + 1 2 ) – 2 +5 = 2(x – 1) 2 + 3. ÕÚ·, Ë ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙË˜ f ÚÔÎ‡ÙÂÈ ·fi ‰‡Ô ‰È·‰Ô¯ÈÎ¤˜ ÌÂÙ·- ÙÔ›ÛÂÈ˜ ÙË˜ ÁÚ·ÊÈÎ‹˜ ·Ú¿ÛÙ·ÛË˜ ÙË˜ g(x) = 2x 2 , ÌÈ·˜ ÔÚÈ˙fiÓÙÈ·˜ Î·Ù¿ 1 ÌÔÓ¿‰· ÚÔ˜ Ù· ‰ÂÍÈ¿ Î·È ÌÈ·˜ Î·Ù·ÎfiÚ˘ÊË˜ Î·Ù¿ 3 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ¿Óˆ. ii) Œ¯Ô˘ÌÂ f(x) = – 2(x 2 – 4x) – 9 = –2(x 2 – 2 Ø 2x + 2 2 ) + 8 – 9 = –2(x – 2) 2 – 1. ÕÚ·, Ë ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙË˜ f ÚÔÎ‡ÙÂÈ ·fi ‰‡Ô ‰È·‰Ô¯ÈÎ¤˜ ÌÂÙ·- ÙÔ›ÛÂÈ˜ ÙË˜ ÁÚ·ÊÈÎ‹˜ ·Ú¿ÛÙ·ÛË˜ ÙË˜ g(x) = –2x 2 , ÌÈ·˜ ÔÚÈ˙fiÓÙÈ·˜ Î·Ù¿ 2 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ‰ÂÍÈ¿ Î·È ÌÈ·˜ Î·Ù·ÎfiÚ˘ÊË˜ Î·Ù¿ 1 ÌÔÓ¿‰· ÚÔ˜ Ù· Î¿Ùˆ. 2. ·) °È· ÙË Û˘Ó¿ÚÙËÛË f(x) = 2x 2 – 6x + 3 Â›Ó·È · = 2 > 0, ÔfiÙÂ ·˘Ù‹ ·- ÚÔ˘ÛÈ¿˙ÂÈ ÂÏ¿¯ÈÛÙÔ ÁÈ· ‚) °È· ÙË Û˘Ó¿ÚÙËÛË g(x) = –3x 2 – 5x + 2 Â›Ó·È · = –3 < 0, ÔfiÙÂ ·˘Ù‹ ·ÚÔ˘ÛÈ¿˙ÂÈ Ì¤ÁÈÛÙÔ ÁÈ· x = – = 6 4 = 3 2 , ÙÔ f 3 2 = 2 3 2 2 – 6 3 2 + 3 = – 3 2 . y = 4 x y = 4 x xy 2 xy 2 1 x ∫∂º∞§∞π√ 7: ª∂§∂Δ∏ μ∞™π∫ø¡ ™À¡∞ƒΔ∏™∂ø¡ 100
3. ·) °È· ÙË Û˘Ó¿ÚÙËÛË f(x) = 2x 2 + 4x + 1 Â›Ó·È · = 2 > 0, ÔfiÙÂ ·˘Ù‹ ¶·ÚÔ˘ÛÈ¿˙ÂÈ ÂÏ¿¯ÈÛÙÔ ÁÈ· ∂›Ó·È ÁÓËÛ›ˆ˜ Êı›ÓÔ˘Û· ÛÙÔ (– , –1] Î·È ÁÓËÛ›ˆ˜ ·‡ÍÔ˘Û· ÛÙÔ [–1, + ). ∞ÎfiÌË Ë ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙË˜ f Â›Ó·È ·Ú·‚ÔÏ‹ Î·È ¤¯ÂÈ ÎÔÚ˘Ê‹ ÙÔ ÛËÌÂ›Ô ∫(–1, –1) Î·È ¿ÍÔÓ· Û˘ÌÌÂÙÚ›·˜ ÙËÓ Â˘ıÂ›· x = –1, Ù¤ÌÓÂÈ ÙÔÓ ¿ÍÔÓ· xãx ÛÙ· ÛËÌÂ›· ÔÈ ÙÂÙÌËÌ¤ÓÂ˜ ÙˆÓ ÔÔ›ˆÓ, Â›Ó·È ÔÈ Ú›˙Â˜ ÙÔ˘ ÙÚÈˆÓ‡ÌÔ˘ 2x 2 + 4x + 1, ÂÓÒ ÙÔÓ ¿ÍÔÓ· yãy ÛÙÔ ÛËÌÂ›Ô °(0, 1).

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