设f(x)=e^x-e^-x,g(x)=e^x+e^-x(e=2.71828)先判断函数f(x)的单调性,再解不等式f(x)>f(-x+2);设f(x)*f(y)=3,g(x)*g(y)=7,求g(x-y)/g(x+y)的值
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![设f(x)=e^x-e^-x,g(x)=e^x+e^-x(e=2.71828)先判断函数f(x)的单调性,再解不等式f(x)>f(-x+2);设f(x)*f(y)=3,g(x)*g(y)=7,求g(x-y)/g(x+y)的值](/uploads/image/z/1058108-68-8.jpg?t=%E8%AE%BEf%28x%29%3De%5Ex-e%5E-x%2Cg%28x%29%3De%5Ex%2Be%5E-x%28e%3D2.71828%29%E5%85%88%E5%88%A4%E6%96%AD%E5%87%BD%E6%95%B0f%28x%29%E7%9A%84%E5%8D%95%E8%B0%83%E6%80%A7%2C%E5%86%8D%E8%A7%A3%E4%B8%8D%E7%AD%89%E5%BC%8Ff%28x%29%26gt%3Bf%28-x%2B2%29%3B%E8%AE%BEf%28x%29%2Af%28y%29%3D3%2Cg%28x%29%2Ag%28y%29%3D7%2C%E6%B1%82g%28x-y%29%2Fg%28x%2By%29%E7%9A%84%E5%80%BC)
设f(x)=e^x-e^-x,g(x)=e^x+e^-x(e=2.71828)先判断函数f(x)的单调性,再解不等式f(x)>f(-x+2);设f(x)*f(y)=3,g(x)*g(y)=7,求g(x-y)/g(x+y)的值
设f(x)=e^x-e^-x,g(x)=e^x+e^-x(e=2.71828)
先判断函数f(x)的单调性,再解不等式f(x)>f(-x+2);
设f(x)*f(y)=3,g(x)*g(y)=7,求g(x-y)/g(x+y)的值
设f(x)=e^x-e^-x,g(x)=e^x+e^-x(e=2.71828)先判断函数f(x)的单调性,再解不等式f(x)>f(-x+2);设f(x)*f(y)=3,g(x)*g(y)=7,求g(x-y)/g(x+y)的值
第一问 两种方法,若用导数,f(x)‘e^x+e^(-x)>0,函数在定义域内单调递增!若普通方法,不妨设x1>x2,f(x1)-f(x2)=e^x1-e^x2+1/e^x2-1/e^x1=(e^x1-e^x2)(1/e^(x1+x2) + 1),后一项永远大于零,前一项因为x1>x2,根据指数函数的图像单调递增,可知e^x1-e^x2>0,所以,f(x1)-f(x2)>0,函数在定义域内单调递增.由上问函数单调递增,则 f(x)>f(-x+2),只需 x>-x+2,即x>1
第二问 f(x)*f(y)=3,则 (e^x-e^-x)(e^y-e^-y)=3,(e^2x-1)/e^x * (e^2y-1)/e^y=[e^2(x+y)-(e^x+e^y)+1]/e^(x+y)=3 g(x)*g(y)=7,同理可得到 [e^2(x+y)+(e^x+e^y)+1]/e^(x+y)=7,两式相加得2(e^2(x+y)+1)/e^(x+y)=10,两式相减得,2(e^2x+e^2y)/e^(x+y)=4,因为g(x-y)/g(x+y)=(e^2x+e^2y)/[e^2(x+y)+1]=[2(e^2x+e^2y)/e^(x+y)]/[2(e^2(x+y)+1)/e^(x+y)]=4/10=2/5