证明下列不等式:
(1)当x>1时,ex>ex;
(2)当x>1时,2√x>3-1/x.
(1)令f(x)=ex-ex,则f(x)的定义域为(-∞,+∞),而 f'(x)=ex-e, 令f''(x)=0,得x=1. 当x>1时,f'(x)>0,故f(x)在(1,+∞)上单调增加,故有 f(x)>f(1),即ex-ex>0, 故ex>ex. (2)令f(x)=2√x+1/x-3,则f(x)的定义域为(0,+∞),在(0,+∞)上, f'(x)=1/√x-1/x2=x2-√x/x2√x, 当x>1时,f'(x)=√x(x√x-1)/x2√x>0,故f(x)在(1,+∞)上为增函数,因此有 f(x)=2√3-(3-1/x)>f(1)=0, 即2√x>3-1/x.