A) \[{{2}^{1/2}}\]
B) \[{{3}^{1/3}}\]
C) \[{{7}^{1/4}}\]
D) all but 1 are equal
Correct Answer: B
Solution :
| 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 \[1<2<e\] \[\therefore \operatorname{Max}\{1,{{2}^{1/2}}\}={{2}^{1/2}}\] But \[{{2}^{1/2}}={{4}^{1/4}}<{{3}^{1/3}}\] |
| \[\therefore \] The greatest number is \[{{3}^{1/3}}\] |
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