• # question_answer The greatest of the numbers  $1,{{2}^{1/2}},{{3}^{1/3}},{{4}^{1/4}},{{5}^{1/5}},{{6}^{1/6}}\operatorname{and}{{7}^{1/7}}$is A) ${{2}^{1/2}}$ B) ${{3}^{1/3}}$ C) ${{7}^{1/4}}$ D) all but 1 are equal

 If $f(x)={{x}^{1/x}}$,  then$f'\left( x \right)=\frac{1}{{{x}^{2}}}\left[ {{x}^{1/x}}\left( 1-\ln x \right) \right]$ f is decreasing if $x>e$and f is increasing if $x>e.$ As $e<3<4<5<6<7$ $\therefore$$Max\{{{3}^{1/3}},{{4}^{1/4}},{{5}^{1/5}},{{6}^{1/6}},{{7}^{1/7}}\}={{3}^{1/3}}$ Also \[1<2