推 Jer1983 :請問第二題 |f'(x)| 會是有界嗎? 02/20 00:08
→ Jer1983 :其實這是李杰高微裡面的一個題目 我懷疑打錯了 02/20 00:08
推 silvermare :f(x)=log(x) 02/20 01:55
※ 引述《Jer1983 (stanley)》之銘言：
: 1.
: let f:[a,b] -> R be a differentialble function. f'(a) = +infinity
: f'(b) = -infinity. For c in R, there exists x and y in [a,b]
: such that f'(x) > c and f'(y) < c. 請問這件事是怎麼做到的?
我想題目應該是there exists x and y in (a,b) such that....
If not, then there exists c such that |f'(x)| < c for all x in (a,b)
By mean value theorem, |(f(x)-f(a))/(x-a)|=|f'(d)|< c for some d in (a,b)
Let x goes to a, then f'(a) is bounded.
: 2.
: let f:(a,b) -> R be a differentiable function, then |f(x)| <= K for
: x in (a,b). 請問這邊是怎麼來的? (我只知道連續函數在閉區間是有界)
這是錯的, f(x)=1/x in (0,1) 就是反例
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