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Question Number 12015 by Nayon last updated on 09/Apr/17

$$provethatforallx\in R,$$$$e^x⩾x^e$$

Answered by mrW1 last updated on 18/Apr/17

$$atx=e,e^x=x^e$$$$f\left(x\right)=e^x$$$$g\left(x\right)=x^e=e^{e\ln x}$$$$f^{\prime }\left(x\right)=e^x$$$$f^{″}\left(x\right)=e^x$$$$g^{\prime }\left(x\right)=e^{e\ln x}\frac{e}{x}$$$$g^{″}\left(x\right)={\left(\frac{e}{x}\right)}^2{e}^{e\ln x}-e^{e\ln x}\frac{e}{x^2}=\frac{e^{x+1}}{x^2}(e-1)$$$$f^{″}\left(e\right)=e^e$$$$g^{″}\left(e\right)=\frac{e^{e+1}}{e^2}(e-1)=e^{e-1}(e-1)<e^{e-1}e=e^e$$$$\Rightarrow f^{″}\left(e\right)>g^{″}\left(e\right)$$$$\Rightarrow f\left(x\right)⩾g\left(x\right)$$

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