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‰Â ‰Â‡ÙÂÚË˜ Î·Ù¿ ‰‡Ô ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ·ÚÈÛÙÂÚ¿. 4. i) ∏ ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙˆÓ f(x) = x 2 Â›Ó·È Ë ·Ú·‚ÔÏ‹ y = x 2 ÙÔ˘ ‰ÈÏ·ÓÔ‡ Û¯‹Ì·ÙÔ˜, ÂÓÒ Ë ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙË˜ g(x) = 1 Â›Ó·È Ë Â˘ıÂ›· y = 1 ÙÔ˘ ›‰ÈÔ˘ Û¯‹Ì·ÙÔ˜. OÈ ÁÚ·ÊÈÎ¤˜ ·Ú·ÛÙ¿ÛÂÈ˜ Ù¤- ÌÓÔÓÙ·È ÛÙ· ÛËÌÂ›· A(1, 1) Î·È B(–1, 1) Ô˘ Â›Ó·È Û˘ÌÌÂÙÚÈÎ¿ ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· yãy. ∂ÂÈ‰‹ x 2 ≤ 1 f(x) ≤ g(x) Î·È x 2 > 1 f(x) > g(x) Ë ·Ó›ÛˆÛË x 2 ≤ 1 ·ÏËıÂ‡ÂÈ ÁÈ· ÂÎÂ›Ó· Ù· x ÁÈ· Ù· ÔÔ›· C f ‚Ú›ÛÎÂÙ·È Î¿Ùˆ ·fi ÙËÓ C g ‹ ¤¯ÂÈ ÙÔ ›‰ÈÔ ‡„Ô˜ ÌÂ ·˘Ù‹, ÂÓÒ Ë x 2 > 1 ·ÏËıÂ‡ÂÈ ÁÈ· ÂÎÂ›Ó· Ù· x ÁÈ· Ù· ÔÔ›· Ë C f ‚Ú›ÛÎÂÙ·È ¿Óˆ ·fi ÙËÓ C g . ∂ÔÌ¤Óˆ˜, ı· ¤¯Ô˘ÌÂ x 2 ≤ 1 –1 ≤ x ≤ 1 Î·È x 2 > 1 x < –1 x > 1. ii) Œ¯Ô˘ÌÂ x 2 ≤ 1 x 2 – 1 ≤ 0 x [–1, 1] x 2 > 1 x 2 – 1 > 0 x (– , –1) (1, + ) ‰ÈfiÙÈ ÙÔ ÙÚÈÒÓ˘ÌÔ x 2 – 1 ¤¯ÂÈ Ú›˙Â˜ ÙÈ˜ x 1 = – 1 Î·È x 2 = 1. ∫∂º∞§∞π√ 7: ª∂§∂Δ∏ μ∞™π∫ø¡ ™À¡∞ƒΔ∏™∂ø¡ 94

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μã √ª∞¢∞™ 1. ∂›Ó·È –x 2 , x < 0 f(x) = x 2 , x ≥ 0 ∂ÔÌ¤Óˆ˜, Ë ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙË˜ f ·ÔÙÂÏÂ›Ù·È ·fi ÙÔ ÙÌ‹Ì· ÙË˜ ·Ú·- ‚ÔÏ‹˜ y = –x 2 ÙÔ˘ ÔÔ›Ô˘ Ù· ÛËÌÂ›· ¤¯Ô˘Ó ·ÚÓËÙÈÎ‹ ÙÂÙÌËÌ¤ÓË Î·È ·fi ÙÔ ÙÌ‹Ì· ÙË˜ ·Ú·‚ÔÏ‹˜ y = x 2 ÙÔ˘ ÔÔ›- Ô˘ Ù· ÛËÌÂ›· ¤¯Ô˘Ó ÙÂÙÌËÌ¤ÓË ıÂÙÈÎ‹ ‹ ÌË‰¤Ó. 2. ∏ ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË˜ ÙË˜ –x, x < 0 f(x) = x 2 , x ≥ 0 ·ÔÙÂÏÂ›Ù·È ·fi ÙÔ ÙÌ‹Ì· ÙË˜ Â˘ıÂ›·˜ y = –x ÙÔ˘ ÔÔ›Ô˘ Ù· ÛËÌÂ›· ¤¯Ô˘Ó ·ÚÓËÙÈÎ‹ ÙÂÙÌËÌ¤ÓË Î·È ·fi ÙÔ ÙÌ‹- Ì· ÙË˜ ·Ú·‚ÔÏ‹˜ y = x 2 ÙÔ˘ ÔÔ›Ô˘ Ù· ÛËÌÂ›· ¤¯Ô˘Ó ÙÂÙÌËÌ¤ÓË ıÂÙÈÎ‹ ‹ ÌË‰¤Ó. ∞fi ÙË ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙË˜ f ÚÔ- ÛÎ‡ÙÂÈ fiÙÈ ∏ f Â›Ó·È ÁÓËÛ›ˆ˜ Êı›ÓÔ˘Û· ÛÙÔ (– , 0] Î·È ÁÓËÛ›ˆ˜ ·‡ÍÔ˘Û· ÛÙÔ [0, + ). ∏ f ·ÚÔ˘ÛÈ¿˙ÂÈ ÂÏ¿¯ÈÛÙÔ ÁÈ· x = 0, ÙÔ f(0) = 0. 3. i) ∞fi ÙÔ Û¯‹Ì· ·˘Ùfi ÚÔÎ‡ÙÂÈ fiÙÈ ·) ™ÙÔ ‰È¿ÛÙËÌ· (0, 1) ·fi fiÏÂ˜ ÙÈ˜ ÁÚ·ÊÈÎ¤˜ ·Ú·ÛÙ¿ÛÂÈ˜ ¯·ÌËÏfiÙÂÚ· ‚Ú›ÛÎÂÙ·È Ë y = x 3 , ¤ÂÈÙ· Ë y = x 2 , ¤ÂÈÙ· Ë y = x Î·È Ù¤ÏÔ˜ Ë ∂ÔÌ¤Óˆ˜, ·Ó x (0, 1) ÙfiÙÂ x 3 < x 2 < x < ‚) ™ÙÔ ‰È¿ÛÙËÌ· (1, + ) Û˘Ì‚·›ÓÂÈ ÙÔ ·ÓÙ›ıÂÙÔ. ∂ÔÌ¤Óˆ˜ ·Ó x (1, + ), ÙfiÙÂ x 3 > x 2 > x > ii) ñ ŒÛÙˆ 0 < x< 1. TfiÙÂ x 3 < x 2 x 2 (x – 1) < 0, Ô˘ ÈÛ¯‡ÂÈ, ‰ÈfiÙÈ 0 < x < 1.
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