The following is proof of where the function x to the x power has its minimum value. The proof involves logarithmic differentiation and the first derivative test from calculus. Note that the proof uses the law of logarithms that converts exponents into factors , i.e. brings the power around to the front.

As you can see from the graph of x raised to the x power, the domain is all positive real numbers and the function achieves an absolute minimum value at 1 over e or 1/e which is approximately 0.37.

Also observe that analytically 1/e is the location of minimum value by the first derivative test.

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