试题分析:设动点P(m,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001950807382.png)
)(m>0),则y
′=-
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001950823392.png)
,∴f
′(m)=-
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001950885449.png)
,
∴过动点P(m,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001950807382.png)
)的切线方程为:y-
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001950807382.png)
=-
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001950885449.png)
(x-m).
①分别令y=0,x=0,得A(2m,0),B(0,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951010445.png)
).
则|PA|=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951026636.png)
,|PB|=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951026636.png)
,∴|PA|=|PB|,故①正确;
②由上面可知:△OAB的周长=2m+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951010445.png)
+2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951026636.png)
≥2×2+2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951104593.png)
=4+2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951104344.png)
,当且仅当m=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001950807382.png)
,即m=1时取等号.故△OAB的周长有最小值4+2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951104344.png)
,即②正确.
③假设曲线C上存在两点M(a,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951150327.png)
),N(b,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951166342.png)
),不妨设0<a<b,∠OMN=90°.
则|ON|=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951104344.png)
|OM|,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824001951197606.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240019512131761.png)
化为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240019512131119.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240019512281172.png)
,故假设成立.因此③正确.
故选C。
点评:理解导数的几何意义、基本不等式的性质、两点间的距离公式及等腰直角三角形的性质是解题的关键.较难。