Think of a number between 2 and 3, like 2.37. Fire up your favorite calculator. Compute
(2.37)^(1/2.37)
where ^ means "raise to the power" and / means "divide."
You should get something like
1.439201705
Challenge: Play with this choice, namely 2.37, so that when you calculate
(choice)^(1/choice)
you obtain the biggest value you can. For example:
(2.62)^(1/2.62)
should give you something like
1.444298571
It turns out, the choice that makes the calculation bigger than any other choice is a very very famous number.
What's your choice? As they say on tests, "show your work."
Extra credit:
What is the famous number? How did you determine this?
1 comment:
Problem solving is a many-to-one undertaking ... one goal, many ways to get there. One thing about this one is the idea that the "function" x^(1/x) is claimed to reach a largest value when x is equal to a particular value so if you could somehow find some "graphing" utility on the internet and ask it to "graph" this "function" you might be able to see its high point and thus zoom in on the value of x that you're looking for.
If you google on "function graphing utility" you'll get a hit on
http://www.walterzorn.com/grapher/grapher_e.htm
The other thing that is worth noting is this: if you click on the calculator image in this post you notice that the the value of x that was tried was
2.713
which has 3 decimal digit accuracy, 1 more digit than the two previously mentioned guesses (2.37 and 2.62). This suggests the problem can be attacked by trying things like
2.713 2.714 2.715 2.716 2.717 2.718 2.719 2.720 etc.
and looking at the results to see if they get bigger or smaller.
And then ... maybe go for 4 decimal digits.
2.7181 2.7182 2.7183 etc.
Finally, googling on one of the "guesstimates," like 2.713 or 2.718 etc. might be fruitful.
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