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# Maximum of nth roots of n (TOMATO Subjective 150)

Find the maximum among $$\mathbf { 1 , 2^{1/2} , 3^{1/3} , 4^{1/4} , … }$$ .

Discussion

Consider the function $$\mathbf { f(x) = x^{1/x} }$$ . We employ standard techniques to compute the maxima.

Take logarithm on both sides we have $$\mathbf { \log f(x) = \frac{1}{x} \log x }$$ . Next find out the derivative:

$$\mathbf {\frac {1}{f(x)} f'(x) = \frac{-1}{x^2} \log x + \frac{1}{x}\cdot\frac{1}{x} implies f'(x) = f(x) \cdot \frac{1}{x^2} (1 – \log x) }$$