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问题 解答题
已知函数f（x）=x-ln（x+1）．
（1）求函数f（x）的最小值；
（2）求证：e1+
1
2
+
1
3
+
1
4
+…+
1
n
＞n+1，（n∈N*）．
答案

（1）f′(x)=1-

1
x+1
=
x
x+1
，（x＞-1）．

令f′（x）＞0，解得x＞0，函数f（x）在（0，+∞）上单调递增；f′（x）＜0，解得-1＜x＜0，函数f（x）单调递减．

∴当x=0时，函数f（x）取得极小值，即最小值f（0）=0；

（2）由（1）可知：x＞0时，f（x）＞0，即x-ln（x+1）＞0，即ln（x+1）＜x．

令x=

1
n
，得ln
n+1
n
＜
1
n
．

∴ln2+ln

3
2
+…+ln
n+1
n
＜1+
1
2
+
1
3
+…+
1
n
，

∴ln(2×

3
2
×…
n+1
n
)=ln（n+1）＜1+
1
2
+
1
3
+…+
1
n
，

∴n+1＜e1+

1
2
+
1
3
+…+
1
n
．

阅读理解

阅读小天地。

      Jim is my friend. He lives in America. There are five people in his family: his father, mother, sister and

brother.

      His father is Mr. Smith. He is forty-one years old. He is a doctor. Jim's mother is a bus driver. She is forty

years old. His sister's name is Rose. She is older than Jim. His brother, Bob is younger than Jim.

      Jim and his brother look different. Jim has black hair and blue eyes, but his brother has brown hair.

根据短文内容，判断下列句子的正(T)误(F)。

( ) 1. Jim is an American.

( ) 2. Jim's father has 3 sons.

( ) 3. Jim's mother is forty-one years old.

( ) 4. Jim's sister is younger than Jim's brother.

( ) 5. Bob has brown hair.

( ) 6. The two boys look the same.

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单项选择题

能量既不会凭空产生，也不会凭空消失，它只能从一种形式转化为另一种形式，或者从 ______一个物体转移到别的物体，在转化或转移的过程中其总量不变，这就是能量守恒定律。如果在一个密封的，即不存在热量的进入或散失的房间内，打开一个正在工作的电冰箱门，最终房间内的温度将______。

A．降低
B．不变
C．升高
D．无法确定
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