The maximum value of x^(1/x) , x > 0 is A. e^(1/e) B. (1/e)^e C.1 D. None of these

Question

The maximum value of x^1/x , x > 0 is :

A. e^1/e 

B. \((\frac{1}{e})^e\)

C.1 

D. None of these

Answer 1

Option : (B)

f(x) = \(x^{\frac{1}{x}}\) 

Let y = \(x^{\frac{1}{x}}\) 

Therefore,

logy = \(log_ex\)

Differentiating w.r.t x,

So,

Now, 

lets put y’ = 0

So,

\(x^{\frac{1-2x}{x}}\) = 0

Or,

(1 - log x) = 0

Therefore,

x = 1/e and x = 0 

Hence by second derivative test 

f’’(1/e) < 0 so it’s a point of maximum and maximum value is \((\frac{1}{e})^e\).