b ____ t ____  ____ fly

已知函数f（x）=lnx﹣x+a有且只有一个零点．

（1）求a的值；

（2）若对任意的x∈（1，+∞），有2f（x）＜$\frac{k}{x}$﹣x+2恒成立，求实数k的最小值；

（3）设h（x）=f（x）+x﹣1，对任意x₁，x₂∈（0，+∞）（x₁≠x₂），证明：不等式$\frac{x_1-x_2}{h\left(x_1\right)-h\left(x_2\right)}$>$\sqrt{x_1{x}_2}$恒成立．

阅读下文，完成小题。

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