è§£ï¼ (1) ç±é¢æå¾ f (e) = peï¼e(q)ï¼2ln e = qeï¼e(p)ï¼2
Þ (pï¼q) (e + e(1)) = 0
è e + e(1)â 0 â´ p = q
(2) ç± (1) ç¥ f (x) = pxï¼x(p)ï¼2ln x
fâ(x) = p + x 2(p)ï¼x(2)= x 2(px 2ï¼2x + p)
è¦ä½¿ f (x) å¨å
¶å®ä¹å (0,+¥) å
为åè°å¢å½æ°ï¼åªé fâ(x) å¨ (0,+¥) å
满足ï¼
fâ(x)â¥0ææç«.
ç± fâ(x)â¥0 Û p (1 + x 2(1))ï¼x(2)â¥0 Û pâ¥x(1) Û pâ¥(x(1))maxï¼x > 0
âµ x(1)⤠x(1)= 1ï¼ä¸ x = 1 æ¶çå·æç«ï¼æ
(x(1))max = 1
â´ pâ¥1
(3) âµ g(x) = x(2e)å¨ [1,e] ä¸æ¯åå½æ°
â´ x = e æ¶ï¼g(x)min = 2ï¼x = 1 æ¶ï¼g(x)max = 2eå³ g(x) Î [2,2e]
â 0 < p < 1 æ¶ï¼ç±x Î [1,e] Þ xï¼x(1)â¥0
â´ f (x) = p (xï¼x(1))ï¼2ln xâ¤xï¼x(1)ï¼2ln x
å½ p = 1 æ¶ï¼f (x)= xï¼x(1)ï¼2ln xå¨ [1,e] éå¢
â´ f (x)â¤xï¼x(1)ï¼2ln xâ¤eï¼e(1)ï¼2ln e = eï¼e(1)ï¼2 < 2ï¼ä¸åé¢æã
â¡ pâ¥1 æ¶ï¼ç± (2) ç¥ f (x) å¨ [1,e] è¿ç»éå¢ï¼f (1) = 0 < 2ï¼åg(x) å¨ [1,e] ä¸æ¯åå½æ°
â´ æ¬å½é¢ Û f (x)max > g(x)min = 2ï¼x Î [1,e]
Þ f (x)max = f (e) = p (eï¼e(1))ï¼2ln e > 2 Þ p > e 2ï¼1(4e)
综ä¸ï¼p çåå¼èå´æ¯ (e 2ï¼1(4e),+¥)
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