1. ## Sort this numbers

Sort the following numbers of smallest to largest
$
\left( {\sqrt 2 } \right)^{\frac{1}{{\sqrt 2 }}} ,\left( {\sqrt e } \right)^{\frac{1}{{\sqrt e }}} ,\left( {\sqrt 3 } \right)^{\frac{1}{{\sqrt 3 }}} ,\left( {\sqrt \pi } \right)^{\frac{1}{{\sqrt \pi }}} ,\left( 2 \right)^{\frac{1}{2}} ,\left( {\sqrt 5 } \right)^{\frac{1}{{\sqrt 5 }}} ,\left( 3 \right)^{\frac{1}{3}} ,\left( 4 \right)^{\frac{1}{4}} ,\left( 5 \right)^{\frac{1}{5}}

$

2. Originally Posted by dhiab
Sort the following numbers of smallest to largest
$
\left( {\sqrt 2 } \right)^{\frac{1}{{\sqrt 2 }}} ,\left( {\sqrt e } \right)^{\frac{1}{{\sqrt e }}} ,\left( {\sqrt 3 } \right)^{\frac{1}{{\sqrt 3 }}} ,\left( {\sqrt \pi } \right)^{\frac{1}{{\sqrt \pi }}} ,\left( 2 \right)^{\frac{1}{2}} ,\left( {\sqrt 5 } \right)^{\frac{1}{{\sqrt 5 }}} ,\left( 3 \right)^{\frac{1}{3}} ,\left( 4 \right)^{\frac{1}{4}} ,\left( 5 \right)^{\frac{1}{5}}

$
Hi

The function $f(x) = x^{\frac{1}{x}} = e^{\frac{\ln x}{x}}$ is differentiable over $]0,+\infty[$ and $f'(x) = \frac{1-\ln x}{x^2}\:e^{\frac{\ln x}{x}}$

Therefore f is increasing over ]0,e] and decreasing over [e,+oo[ and
$\left( {\sqrt 2 } \right)^{\frac{1}{{\sqrt 2 }}} < \left( {\sqrt e } \right)^{\frac{1}{{\sqrt e }}} < \left( {\sqrt 3 } \right)^{\frac{1}{{\sqrt 3 }}} < \left( {\sqrt \pi } \right)^{\frac{1}{{\sqrt \pi }}} < \left( 2 \right)^{\frac{1}{2}} < \left( {\sqrt 5 } \right)^{\frac{1}{{\sqrt 5 }}}$

and $\left( 5 \right)^{\frac{1}{5}} < \left( 4 \right)^{\frac{1}{4}} < \left( 3 \right)^{\frac{1}{3}}$

Then you can use the fact that $\left( 4 \right)^{\frac{1}{4}} = \left(\left( 4 \right)^{\frac{1}{2}}\right)^{\frac{1}{2}} = \left( 2 \right)^{\frac{1}{2}}$

3. Originally Posted by running-gag
Hi

The function $f(x) = x^{\frac{1}{x}} = e^{\frac{\ln x}{x}}$ is differentiable over $]0,+\infty[$ and $f'(x) = \frac{1-\ln x}{x^2}\:e^{\frac{\ln x}{x}}$

Therefore f is increasing over ]0,e] and decreasing over [e,+oo[ and
$\left( {\sqrt 2 } \right)^{\frac{1}{{\sqrt 2 }}} < \left( {\sqrt e } \right)^{\frac{1}{{\sqrt e }}} < \left( {\sqrt 3 } \right)^{\frac{1}{{\sqrt 3 }}} < \left( {\sqrt \pi } \right)^{\frac{1}{{\sqrt \pi }}} < \left( 2 \right)^{\frac{1}{2}} < \left( {\sqrt 5 } \right)^{\frac{1}{{\sqrt 5 }}}$

and $\left( 5 \right)^{\frac{1}{5}} < \left( 4 \right)^{\frac{1}{4}} < \left( 3 \right)^{\frac{1}{3}}$

Then you can use the fact that $\left( 4 \right)^{\frac{1}{4}} = \left(\left( 4 \right)^{\frac{1}{2}}\right)^{\frac{1}{2}} = \left( 2 \right)^{\frac{1}{2}}$
Hello thankyou : How do you compare two numbers:

$\left( {\sqrt 2 } \right)^{\frac{1}{{\sqrt 2 }}} ,\left( {\sqrt 5 } \right)^{\frac{1}{{\sqrt 5 }}}$Thank you