what-is-larger-99-100-or-100-99-

Question Number 137473 by mr W last updated on 03/Apr/21

w h a t i s l a r g e r ? 99^{100} o r 100^{99} ?

Answered by MJS_new last updated on 03/Apr/21

n^{n + 1} > {(n + 1)}^{n} \forall n ⩾ 3

Commented by mr W last updated on 03/Apr/21

y e s s i r .

Answered by mr W last updated on 03/Apr/21

f (x) = x^{\frac{1}{x}} = e^{\frac{\ln x}{x}}

f^{'} (x) = x^{\frac{1}{x}} (\frac{1}{x^{2}} - \frac{\ln x}{x^{2}}) = (1 - \ln x) x^{\frac{1}{x} - 2}

f o r x < e : f^{'} (x) > 0 \Rightarrow f (x) i s s t r i c t l y i n c r e a s i n g

f o r x > e : f^{'} (x) < 0 \Rightarrow f (x) i s s t r i c t l y d e c r e a s i n g

t h a t m e a n s f o r 3 ⩽ m < n :

m^{\frac{1}{m}} > n^{\frac{1}{n}}

\Rightarrow {(m^{\frac{1}{m}})}^{m n} > {(n^{\frac{1}{n}})}^{m n}

\Rightarrow m^{n} > n^{m}

s o 99^{100} > 100^{99}

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