<s>[INST] A function inequality about $e^x$ and $\ln x$ If for any $x \in (1,+\infty)$，there is the inequality:
$$x^{-3} e^{x}-a \ln x \geq x+1$$
Find the value range of $a$ .
And I tried constructing the function $f(x)=x^{-3} e^{x}-a \ln x - x-1$ and deriving it, but to no avail.
 [/INST] $$\max{a}=\min_{x>1}\frac{\frac{e^x}{x^3}-x-1}{\ln{x}}.$$
Let $g(x)=\frac{\frac{e^x}{x^3}-x-1}{\ln{x}}.$
Thus, for $e^x=x^3$ or $x=3\ln{x}$ we obtain: $$g'(x)=\frac{x^4+x^3+e^x(x\ln{x}-3\ln{x}-1)-x^4\ln{x}}{x^4\ln^2x}=$$
$$=\frac{x^4+x^3+x^3(x\ln{x}-3\ln{x}-1)-x^4\ln{x}}{x^4\ln^2x}=\frac{x^4-3x^3\ln{x}}{x^4\ln^2x}=0.$$
Also, for $e^x=x^3$ we obtain: $$\frac{\frac{e^x}{x^3}-x-1}{\ln{x}}=-\frac{x
}{\ln{x}}=-3.$$
Now, we'll prove that $$\frac{\frac{e^x}{x^3}-x-1}{\ln{x}}\geq-3$$ or $$\frac{e^x}{x^3}-x-1+3\ln{x}\geq0,$$ where the minimum occurs for $x=x_0$, where $x_0>1$ is a root of the equation:   $e^x=x^3$.
Indeed, let $h(x)=\frac{e^x}{x^3}-x-1+3\ln{x}.$
Thus, $$h'(x)=\frac{e^x}{x^3}-\frac{3e^x}{x^4}-1+\frac{3}{x}=\frac{(x-3)(e^x-x^3)}{x^4}.$$
Now, the equation $e^x=x^3$ or $x=3\ln{x}$ has two roots maximum because $\ln$ is a concave function.
But $3\ln3-3>0$, $3\ln1-1<$ and $3\ln5-5<0$ which says that the equation $e^x=x^3$ has two roots $1<x_0<3$ and $x_1>3$ exactly and easy to see that $x_{max}=3$ and $x_{min_1}=x_0$, $x_{min_2}=x_1.$
Thus, $$h(x)\geq h(x_0)=h(x_1)=0$$ and we got the answer:
$$(-\infty,-3]$$
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