推 halfwater : 謝謝你的幫忙！~~~ 03/15 00:06
※ 引述《halfwater (半瓶水)》之銘言：
: 如圖，謝謝 ~~~
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1. 由於 g(x) 是一次函數, 可以由 g(1) 和 g(-1) 的值夾出其範圍
容易知道對 x≠0, g(x) = (f(x) - c)/x = (f(x) - f(0))/x
由此可以寫出 g(1) = (f(1) - f(0))/1 = f(1) - f(0)
g(-1) = (f(-1) - f(0))/(-1) = f(0) - f(-1)
又由題設, 對 -1≦x≦1 有 -5≦f(x)≦5
也就是 -5≦f(-1), f(0), f(1)≦5
於是 -10 = (-5)-(5)≦f(1)-f(0)=g(1)≦(5)-(-5) = 10
-10 = (-5)-(5)≦f(0)-f(-1)=g(-1)≦(5)-(-5) = 10
由此即可得到對 -1≦x≦1 都有 -10≦g(x)≦10
2. 現知最大值為 10, 同樣由於一次函數的性質, 只有兩種可能:
(A) g(1) = 10: 這表示 f(1) = 5, f(0) = -5
注意到 f(0) 已經碰到 f 的範圍最底了, 因此由 0 往左 f 必須要再變大
也就是 (0,-5) 是 f 的頂點, f 凹向上
所以 f(x) = ax^2-5 再代 f(1)=5 得 a=10 也就是 f(x) = 10x^2-5
(B) g(-1) = 10: 對稱的狀況, f(0) = 5, f(-1) = -5
類似地 (0,5) 是 f 的頂點, f 凹向下, 同樣可求得 f(x) = -10x^2+5
--
'Oh, Harry, don't you see?' Hermione breathed. 'If she could have done
one thing to make absolutely sure that every single person in this school
will read your interview, it was banning it!'
---'Harry Potter and the order of the phoenix', P513
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