2011 Λύσεις Σχ. β&I

4 i áúêèî úûùûë ùˆó fx x 2 âóè ë

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‰Â ‰Â‡ÙÂÚ˘ ηٿ ‰‡Ô ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ·ÚÈÛÙÂÚ¿. 4. i) ∏ ÁÚ·ÊÈ΋ ·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·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ù˘ g(x) = 1 Â›Ó·È Ë Â˘ı›· y = 1 ÙÔ˘ ›‰ÈÔ˘ Û¯‹Ì·ÙÔ˜. OÈ ÁÚ·ÊÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ Ù¤- ÌÓÔÓÙ·È ÛÙ· ÛËÌ›· A(1, 1) Î·È B(–1, 1) Ô˘ Â›Ó·È Û˘ÌÌÂÙÚÈο ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· yãy. ∂Âȉ‹ x 2 ≤ 1 f(x) ≤ g(x) Î·È x 2 > 1 f(x) > g(x) Ë ·Ó›ÛˆÛË x 2 ≤ 1 ·ÏËı‡ÂÈ ÁÈ· ÂΛӷ Ù· x ÁÈ· Ù· ÔÔ›· C f ‚Ú›ÛÎÂÙ·È Î¿Ùˆ ·fi ÙËÓ C g ‹ ¤¯ÂÈ ÙÔ ›‰ÈÔ ‡„Ô˜ Ì ·˘Ù‹, ÂÓÒ Ë x 2 > 1 ·ÏËı‡ÂÈ ÁÈ· ÂΛӷ Ù· x ÁÈ· Ù· ÔÔ›· Ë C f ‚Ú›ÛÎÂÙ·È ¿Óˆ ·fi ÙËÓ C g . ∂Ô̤ӈ˜, ı· ¤¯Ô˘Ì x 2 ≤ 1 –1 ≤ x ≤ 1 Î·È x 2 > 1 x < –1 x > 1. ii) Œ¯Ô˘Ì 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›˙˜ ÙȘ 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 ∂Ô̤ӈ˜, Ë ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ù˘ f ·ÔÙÂÏÂ›Ù·È ·fi ÙÔ ÙÌ‹Ì· Ù˘ ·Ú·- ‚ÔÏ‹˜ y = –x 2 ÙÔ˘ ÔÔ›Ô˘ Ù· ÛËÌ›· ¤¯Ô˘Ó ·ÚÓËÙÈ΋ ÙÂÙÌË̤ÓË Î·È ·fi ÙÔ ÙÌ‹Ì· Ù˘ ·Ú·‚ÔÏ‹˜ y = x 2 ÙÔ˘ ÔÔ›- Ô˘ Ù· ÛËÌ›· ¤¯Ô˘Ó ÙÂÙÌË̤ÓË ıÂÙÈ΋ ‹ Ìˉ¤Ó. 2. ∏ ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·Û˘ Ù˘ –x, x < 0 f(x) = x 2 , x ≥ 0 ·ÔÙÂÏÂ›Ù·È ·fi ÙÔ ÙÌ‹Ì· Ù˘ ¢ı›·˜ y = –x ÙÔ˘ ÔÔ›Ô˘ Ù· ÛËÌ›· ¤¯Ô˘Ó ·ÚÓËÙÈ΋ ÙÂÙÌË̤ÓË Î·È ·fi ÙÔ ÙÌ‹- Ì· Ù˘ ·Ú·‚ÔÏ‹˜ y = x 2 ÙÔ˘ ÔÔ›Ô˘ Ù· ÛËÌ›· ¤¯Ô˘Ó ÙÂÙÌË̤ÓË ıÂÙÈ΋ ‹ Ìˉ¤Ó. ∞fi ÙË ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ù˘ f ÚÔ- Û·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 ÚÔ·ÙÂÈ fiÙÈ ·) ™ÙÔ ‰È¿ÛÙËÌ· (0, 1) ·fi fiϘ ÙȘ ÁÚ·ÊÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ ¯·ÌËÏfiÙÂÚ· ‚Ú›ÛÎÂÙ·È Ë y = x 3 , ¤ÂÈÙ· Ë y = x 2 , ¤ÂÈÙ· Ë y = x Î·È Ù¤ÏÔ˜ Ë ∂Ô̤ӈ˜, ·Ó x (0, 1) ÙfiÙ x 3 < x 2 < x < ‚) ™ÙÔ ‰È¿ÛÙËÌ· (1, + ) Û˘Ì‚·›ÓÂÈ ÙÔ ·ÓÙ›ıÂÙÔ. ∂Ô̤ӈ˜ ·Ó x (1, + ), ÙfiÙ x 3 > x 2 > x > ii) ñ ŒÛÙˆ 0 < x< 1. TfiÙ x 3 < x 2 x 2 (x – 1) < 0, Ô˘ ÈÛ¯‡ÂÈ, ‰ÈfiÙÈ 0 < x < 1.
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