If y = xth root of x, why does y reach its maximum value when x = e?

Question

I am not a mathematician but have an interest in maths which doesn't actually go as far as actually doing calculations! I discovered this "fact" by playing with a calculator. I was unable to work out how to differentiate the function which is what you need to do no doubt to show what its maximum is.Can anyone give me some insight as to why this fact is a fact? The one who gets the 10 points will have to explain it in a very simple clear way suitable for someone whose knowledge of analysis is very elementary indeed.

Answer 1

y=x^(1/x)
Take ln
lny = 1/x*ln(x)
now derivate
y´/y= 1/x^2*( 1-ln x)
now y´=0 1-ln x = 0 ln x = 1 and x= e
the sign of y´ is 0++++++ e------- so at e there is a relative max.

Answer 2

hi, what's the smallest useful fee for x the place y = sin 2x reaches its optimal. The era of the sin is 2pi, this graph has 2 cycles in a classic era so it has a era of 2pi/2 = pi. Now the 1st max fee of a classic sin curve is at pi/2 or ninety tiers. on condition that our graph has 0.5 the era of the conventional graph we prefer the cost to be pi/2 divided via 2 or pi/4. it relatively is not one among your options. Why? If we try this with calculus via making use of the by-product we've y' = 2cos2x and this could be a max while it = 0 so we've 0 = 2 cos 2x or dividing via 2 provides us 0 = cos2x now the cos = 0 at pi/2, yet we've 2x = pi/2 so dividing the two sides via 2 provides us x = pi/4. the comparable answer as our first technique. ought to it relatively is that they have not got the excellent answer?? wish This enables!