每日导数86

已知函数\(f(x)=\dfrac{\ln x+1}{x}.g(x)=\dfrac{e^x}{x}\)

(1)若对任意\(m,n\in\left(0,+\infty\right)\)都有\(f(m)\leq t\leq g(n)\)，求实数\(t\)的取值范围.

(2)若对任意\(x_1,x_2\in(0,+\infty)\)且\(x_1\neq x_2,e^{x_2-x_1}=\dfrac{x_2^{x_1}}{x_1^{x_2}}\)，证明\(x_1^3+x_2^3>2\)

解

(1)\(f(x)\leq \dfrac{x-1+1}{x}=1,g(n)\geq \dfrac{ex}{x}=e\)

则\(1\leq t\leq e\)

(2)\(e^{x_2-x_1}=\dfrac{x_2^{x_1}}{x_1^{x_2}}\)取对数有

\(x_2-x_1=x_1\ln x_2-x_2\ln x_1\)，整理得\(\dfrac{1+\ln x_1}{x_1}=\dfrac{1+\ln x_2}{x_1}\)

即\(f(x_1)=f(x_2)\)，即\(x_1<1<x_2\)

现在说明\(x_1+x_2>2\)(极值点偏移做多了，应该想到的)

即说明\(x_2>2-x_1\)

构造\(h(t)=f(t)-f(2-t),t<1\)

即\(h^{\prime}(t)=-\dfrac{\ln x}{x}-\dfrac{\ln(2-x)}{(2-x)^2}=-\dfrac{\ln\left[(-(x-1)^2)+1\right]}{x^2}>0\)

则\(h(t)\)单调递增，则\(h(t)<h(1)=0\)

则\(x_2>2-x_1\)即\(x_1+x_2>2\)

则\(x_1^3+x_2^3>x_1^3+(2-x_1)^3=8-12x_1+6x-1^2=6(x_1-1)^2+2>2\)