# find the maximum number among them

find the maximum number among them::: 1,2^(1/2),3^(1/3),4^(1/4),5^(1/5)........and so on

Note by Sayan Chaudhuri
5 years, 4 months ago

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Consider the function f(x)=x^1/x. It has maxima at x=e. and it is decreasing as we go along either direction away from x=e. The closest integers are 2,3. testing them would be sufficient. Which is greater 2^1/2 or 3^1/3? Simple powering ..L.C.M of 2 and 3 is 6. Powering them by 6 would give 8 and 9. So clearly 3^1/3 is larger.

- 5 years, 4 months ago

The derivative of $$x^{\frac{1}{x}}$$ is $$x^{\frac{1}{x}-2}(1- ln(x))$$. Since this is undefined at 0, the derivative is zero when $$1=ln(x) \Rightarrow x=e$$. So the only critical point is at $$e$$ and it is easy to check it is a maximum value.

- 5 years, 4 months ago

3^(1/3) is the maximum.

- 5 years, 4 months ago