推 raymond168:感謝L大的回答!! 02/05 22:55
※ 引述《raymond168 (raymond168)》之銘言：
: 附上連結:
: http://ppt.cc/8f(T
: 想問的題號分別為:1、2、3(b)、4
: 其中3(b)為第3題的b小題
: 拜託板上各位高手幫忙解惑
: 謝謝
1. Suppose h(t) is a nondecreasing continuous function for t ≧ 0 .
t H(t)
Let H(t) = ∫ h(x) dx . Is ---- nondecreasing for t > 0 ？
0 t
Justify your answer . (10%)
t
sol: H(t) = ∫ h(x) dx => H(0) = 0
0
∵ h(t) is a nondecreasing continuous function for t ≧ 0
t
∴ H(t) = ∫ h(x) dx is a differentiable function for t ≧ 0
0
=> H(t) is a continous function for t ≧ 0
∵ H is continous on [0,t] and differentiable on (0,t) for t ≧ 0
∴ by MVT , there exists c belonging to (0,t) such that
H(t) - H(0) = (H'(c))(t - 0) => H(t) = (t)(h(c)) for 0 < c < t
∵ h(t) is a nondecreasing continuous function for t ≧ 0
∴ h(c) ≦ h(t) for 0 < c < t
=> (t)(h(c)) ≦ (t)(h(t)) => H(t) ≦ (t)(h(t)) -------(＊）
H(t)
Let g(t) = ------
t
(H'(t))(t) - H(t)
Then g'(t) = -------------------
t^2
(t)(h(t)) - H(t)
= ------------------ ≧ 0 for t > 0 (From (＊))
t^2
H(t)
∴ g(t) = ------ is nondecreasing for t > 0
t
--
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