prove-that-for-all-x-R-e-x-x-e-

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

p r o v e t h a t f o r a l l x \in R,

e^{x} ⩾ x^{e}

Answered by mrW1 last updated on 18/Apr/17

a t x = e, e^{x} = x^{e}

f (x) = e^{x}

g (x) = x^{e} = e^{e \ln x}

f^{'} (x) = e^{x}

f^{″} (x) = e^{x}

g^{'} (x) = e^{e \ln x} \frac{e}{x}

g^{″} (x) = {(\frac{e}{x})}^{2} e^{e \ln x} - e^{e \ln x} \frac{e}{x^{2}} = \frac{e^{x + 1}}{x^{2}} (e - 1)

f^{″} (e) = e^{e}

g^{″} (e) = \frac{e^{e + 1}}{e^{2}} (e - 1) = e^{e - 1} (e - 1) < e^{e - 1} e = e^{e}

\Rightarrow f^{″} (e) > g^{″} (e)

\Rightarrow f (x) ⩾ g (x)

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