已知函数f(x)=
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当x≤0时，f（x）＞kx，即x2+1＞kx，x=0时k∈R；
当x＜0时，有x+
＜k，而x+1 x
≤-2（x=-1取等号），所以k＞-2；1 x
故x≤0时，f（x）＞kx恒成立，得k＞-2；
当x＞0时，f（x）＞kx，即为ln（x+1）＞kx，而ln（x+1）＞0，
结合图象可知，要使该不等式恒成立，只需k≤0；
综上，要使f（x）＞kx对任意的x∈R恒成立，k的范围为-2＜k≤0．
故答案为：（-2，0]
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| The famous British inventor George Stephenson was born in 1781 and died in 1848. One of his most important inventions was the train. He 1 his first train when he was forty-four years old. When he was experimenting with the steam engine (蒸汽机)on the train, he met the 2 from other fields, the newspaper journalists and the gentlemen in the country. They said that the noise and the smoke would 3 cows, horses and sheep, that the engine would burst (爆炸) and that hot coals from it would set fire to their houses. At that time, 4 people believed what they said. George Stephenson 5 the people that the train could go on small 6 , and could pull carriages of goods and passengers but there was no great danger to them. It was a very 7 matter for him to make them believe. However, after 8 he was able to do it and the first train that was driven by Stephenson himself 9 what he had said. The first day when the people along the way heard the noises of the train in the distance and saw it 10 quickly to them, they ran back home as quickly as they could and closed their doors for they thought of it as a monster (妖怪). They did not dare to come out until it had passed. | ||||
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