When is the function x^x increasing and decreasing?

Question

If the domain of x*x is all real numbers except zero, how do you find out when the functin is increasing or decreasin? Do I take the derivative and then set the derivative equal to zero?

Please help!

Answer 1

x^x = exp (x ln x)  

we can say that ln is increasing and x is increasing and exp is increasing so x ln x is increasing and also exp(x ln x) is increasing by composition

but for the domain it is not all real numbers except zero 

it is all positive real numbers except zero and for negative integers

for negative numbers it is defined for some rational n/p 

eg  (−1/3)^( −1/3) is not the same as (−2/6)^(−2/6).

so to avoid problem it is defined for all positive real numbers except zero