I’ve recently seen a video where someone did an explanation on why 0 to the power of 0 is defined as 1. He went with X to the power of X and made X increasingly smaller decimals. X = 0.9, X = 0.8, and so on. The results for 0.9 to the power of 0.9 and then 0.8 to the power of 0.8 kept getting smaller until about X = 0.4, where it started to increase again, so 0.3 to the power of 0.3 was a bigger number than 0.4 to 0.4. At what number is the exact “turning point” in this kind of series and why is it in such a weird place (as opposed to at 0.5, i.e halfway to 0)?
In: 5
To find a local minimum one way is to find where the derivative of the function is zero.
the derivate of x^x is simple to to find by using a^b = e^(a ln (b)) so x^x = e^(x ln (x)) let chang that to e^f(x) and f(x)= x ln (x)
The derivate of f ln(x) = 1/x so the derivate f(x) = x/x + 1 ln(x) =1+ ln(x)
The derivate of e^f(x) = f(x)’ e^f(x)
This mean the derivate of x^x = (1+ln(x)) x^x = x^x + ln(x) x^x
The mean point we look for is where ln(x) =-1
e^ln(x) =x it we use that we get e^ln(x)= e^-1 => x= e^-1 =1/e
So the exact point the change happened at 1/e is approximately 0.367879441
Because of the relationship between the power function and the constant e it is not surprising that the answer includes it.
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