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This Questions seems daunting but let’s put our thinking in order.
So how do we approach this Q? one way is to have it as a function of (x) only and express it as
g(x) = x^x + (1-x)^(1-x) and then try to find the f’(x) =0 and solve the usual way… and you can see you’re headed to trouble. So let’s try another way!