2011 Λύσεις Σχ. β&I

Ñ è ùëó g úâè ó âíôìâ fiùè áè

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ñ °È· ÙËÓ g Ú¤ÂÈ Ó· ‰Â›ÍÔ˘Ì fiÙÈ ÁÈ· οı x ÈÛ¯‡Ô˘Ó ÔÈ ·ÓÈÛfiÙË- Ù˜ g(x) ≤ 2 Î·È g(x) ≥ –2. Œ¯Ô˘Ì g(x) ≤ 2 ≤ 2 2x ≤ x 2 + 1 (x – 1) 2 ≥ 0 Ô˘ ÈÛ¯‡ÂÈ Î·È g(x) ≥ –2 ≥ –2 2x ≥ –x 2 – 1 (x + 1) 2 ≥ 0 Ô˘ ÈÛ¯‡ÂÈ. ñ °È· ÙËÓ h Ú¤ÂÈ Ó· ‰Â›ÍÔ˘Ì fiÙÈ ÁÈ· οı x ÈÛ¯‡ÂÈ h(x) ≥ 0 Î·È h(x) ≤ 2. Œ¯Ô˘Ì h(x) ≥ 0 ≥ 0 Ô˘ Â›Ó·È Ê·ÓÂÚfi fiÙÈ ÈÛ¯‡ÂÈ Î·È h(x) ≤ 2 ≤ 2 2x 2 ≤ x 4 + 1 x 4 – 2x 2 + 1 > 0 (x 2 – 1) 2 ≥ 0, Ô˘ ÈÛ¯‡ÂÈ 14. A) i) ¶Ú¤ÂÈ x ≥ 0. ÕÚ· ∞ = [0, + ). ii) AÊÔ‡ ÙÔ ª(·, ‚) ·Ó‹ÎÂÈ ÛÙË ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ù˘ f ¤¯Ô˘Ì 2 = ·. (1) °È· Ó· ·Ó‹ÎÂÈ ÙÔ ªã(‚, ·) ÛÙË ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ù˘ g, Ú¤- ÂÈ g(‚) = · 2 = · Ô˘ ÈÛ¯‡ÂÈ. iii) ∂Âȉ‹ Ù· ÛËÌ›· ª(·, ‚) Î·È ªã(‚, ·) Â›Ó·È Û˘ÌÌÂÙÚÈο ˆ˜ ÚÔ˜ ‚ = · 4x 2 x 4 + 1 4x 2 x 4 + 1 4x x 2 + 1 4x x 2 + 1 2 x 2 + 1 A™∫∏™∂π™ °π∞ ∂¶∞¡∞§∏æ∏ 112
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ÙË ‰È¯ÔÙfiÌÔ Ù˘ 1 ˘ Î·È 3 ˘ ÁˆÓ›·˜ ÙˆÓ ·ÍfiÓˆÓ Û˘ÌÂÚ·›ÓÔ˘Ì fiÙÈ Ë ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ù˘ f(x) = Â›Ó·È Ë Û˘ÌÌÂÙÚÈ΋ Ù˘ ÁÚ·- ÊÈ΋˜ Ù˘ g(x) = x 2 ˆ˜ ÚÔ˜ ÙËÓ Â˘ı›· y = x ÁÈ· x ≥ 0. Afi ÙË ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ù˘ f ÚÔ·ÙÂÈ fiÙÈ Ë f(x) = Â›Ó·È ÁÓËÛ›ˆ˜ ·‡ÍÔ˘Û· ÛÙÔ ∞ = [0, + ) Î·È ¤¯ÂÈ ÔÏÈÎfi ÂÏ¿¯ÈÛÙÔ ÁÈ· x = 0 ÙÔ f(0) = 0. μ) ΔÔ ‰›Ô ÔÚÈÛÌÔ‡ Ù˘ h Â›Ó·È fiÏÔ ÙÔ . Œ¯Ô˘Ì h(–x) = = h(x). ÕÚ· Ë h Â›Ó·È ¿ÚÙÈ· Î·È Ë ÁÚ·ÊÈ΋ Ù˘ ·Ú¿ÛÙ·ÛË ·ÔÙÂÏÂ›Ù·È ·fi ÙË ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ù˘ f Î·È ÙË Û˘ÌÌÂÙÚÈ΋ Ù˘ ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· yãy. °) ™ÙÔ Ù˘¯·›Ô ÙÚ›ÁˆÓÔ ¡ªã¡ã ¤¯Ô˘Ì (¡¡ã) = f(v + 1) = Î·È (¡ªã) = = ÕÚ· ÙÔ ÙÚ›ÁˆÓÔ ¡ªã¡ã Â›Ó·È ÈÛÔÛÎÂϤ˜. 15. ™ÙÔ Î·Ù·ÎfiÚ˘ÊÔ Â›Â‰Ô Ù˘ Á¤Ê˘Ú·˜ ıˆÚԇ̠¤Ó· Û‡ÛÙËÌ· Û˘ÓÙÂÙ·Á- ̤ӈÓ, ÛÙÔ ÔÔ›Ô ·›ÚÓÔ˘Ì ˆ˜ ¿ÍÔÓ· ÙˆÓ x ÙË ¯ÔÚ‰‹ ÙÔ˘ ·Ú·‚ÔÏÈÎÔ‡ ÙfiÍÔ˘ Î·È ˆ˜ ¿ÍÔÓ· ÙˆÓ y ÙË ÌÂÛÔοıÂÙÔ ·˘Ù‹˜ (Û¯‹Ì·). 1 + v = (NNã). (¡ª) 2 + (ªªã) 2 = (v + 1 – v) 2 + ( v) 2 v + 1 |–x| = |x| x x A™∫∏™∂π™ °π∞ ∂¶∞¡∞§∏æ∏ 113
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™ÙÔ Û‡ÛÙËÌ· ·˘Ùfi ÙÔ ·- Ú·‚ÔÏÈÎfi ÙfiÍÔ ¤¯ÂÈ Â͛ۈ- ÛË Ù˘ ÌÔÚÊ‹˜ y = ·x 2 + Á, Ì y ≥ 0 Î·È Ë ÎÔÚ˘Ê‹ ÙÔ˘ Â›Ó·È ÙÔ ÛËÌÂ›Ô ∫(0, 5, 6). ™˘ÓÂÒ˜, Ë Â͛ۈÛË ÙÔ˘ ·Ú·‚ÔÏÈÎÔ‡ ÙfiÍÔ˘ ·›ÚÓÂÈ ÙË ÌÔÚÊ‹ y = ·x 2 + 5,6, Ì y ≥ 0 (1)
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