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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
4 years, 3 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. Balaji Dodda · 4 years, 3 months ago

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@Balaji Dodda 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. Yong See Foo · 4 years, 3 months ago

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3^(1/3) is the maximum. Balaji Dodda · 4 years, 3 months ago

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