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Question Number 205849 by Davidtim last updated on 31/Mar/24

i f a^{a} = b^{b}; a = b i s i t t r u e ?

i f i t i s t r u e t h e n p r o v e i t .

Commented by mr W last updated on 31/Mar/24

n o t t r u e f o r 0 < a, b < 1 .

e x a m p l e :

a, b = \frac{\ln 0.8}{W (\ln 0.8)} \approx {\begin{cases} 0.094649710865 \\ 0.739533650011 \end{cases}

0 . 094649710865^{0.094649710865}

= 0 . 739533650011^{0.739533650011}

= 0.8

Commented by Davidtim last updated on 01/Apr/24

p l e a s e c h e c k t h e a b o v e e x p .

Commented by Davidtim last updated on 01/Apr/24

Commented by mr W last updated on 01/Apr/24

i f {(\frac{x}{3})}^{\frac{x}{3}} = 3^{3} \Rightarrow \frac{x}{3} = 3 \Rightarrow x = 9

t h i s i s t r u e, s i n c e 3^{3} > 1 .

b u t

i f {(\frac{x}{3})}^{\frac{x}{3}} = {(0.8)}^{0.8} \Rightarrow \frac{x}{3} = 0.8 \Rightarrow x = 2.4

t h i s i s n o t v e r y t r u e! s i n c e 1 / \sqrt[e]{e} < 0 . 8^{0.8} < 1 .

t h e c o r r e c t s o l u t i o n i s :

{(\frac{x}{3})}^{\frac{x}{3}} = {(0.8)}^{0.8}

\Rightarrow \frac{x}{3} = \frac{0.8 \ln 0.8}{W (0.8 \ln 0.8)}

\Rightarrow x = \frac{2.4 \ln 0.8}{W (0.8 \ln 0.8)} = {\begin{cases} \approx \frac{2.4 \ln 0.8}{- 2.72587187} = 0.196467 \\ \approx \frac{2.4 \ln 0.8}{- 0.22314355} = 2.4 \end{cases}

Commented by mr W last updated on 01/Apr/24

Commented by mr W last updated on 01/Apr/24

i f x^{x} ⩾ 1 \Rightarrow t h e r e i s o n l y o n e s o l u t i o n f o r x .

i f \frac{1}{\sqrt[e]{e}} < x^{x} < 1 \Rightarrow t h e r e a r e t w o s o l u t i o n s f o r x .

i f x^{x} = \frac{1}{\sqrt[e]{e}} \Rightarrow t h e r e i s o n l y o n e s o l u t i o n f o r x .

i f x^{x} < \frac{1}{\sqrt[e]{e}} \Rightarrow t h e r e i s n o s o l u t i o n f o r x .

Commented by Davidtim last updated on 01/Apr/24

t h a n k s y o u a r e a n e x c e p t i o n a l o f

s c i e n c e .

w o u l d y o u m i n d t o p r o v e t h e a l l o f

t h r e e f o r m s ?

Commented by mr W last updated on 01/Apr/24

y = x^{x} = e^{x \ln x} > 0

y^{'} = e^{x \ln x} (\ln x + 1) = x^{x} (\ln x + 1)

y^{'} = 0 : \ln x + 1 = 0 \Rightarrow x = \frac{1}{e}

t h a t m e a n s :

i f x > \frac{1}{e}, y^{'} > 0 \Rightarrow y = x^{x} i s s t r i c t l y i n c r e a s i n g

i f 0 < x < \frac{1}{e}, y^{'} < 0 \Rightarrow y = x^{x} i s s t r i c t l y d e c r e a s i n g

a t x = \frac{1}{e}, y_{m i n} = \frac{1}{\sqrt[e]{e}}

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