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Mathematics 26 Online
OpenStudy (anonymous):

Please help me generous math wizards!!! Find the coordinates of the two points of intersection of the functions y=x^5 and y=5^x.

OpenStudy (jamesj):

Well obviously x = 5 is one of them, right?

OpenStudy (jamesj):

There other one is more subtle.

OpenStudy (anonymous):

5=x is that right

OpenStudy (anonymous):

neither of those are right lol... Think about the graphs of each...

OpenStudy (jamesj):

x = 5 is most empathetically correct. If x = 5, then y1 = x^5 = 5^5 and y2 = 5^x = 5^5 hence y1 = y2

OpenStudy (anonymous):

okay, and the other point of intersection?

OpenStudy (jamesj):

Is this a quiz for us? Or a question for you?

OpenStudy (anonymous):

The question asked for the coordinates of both points of intersection. Does anybody know the answer?

OpenStudy (anonymous):

I still need to know this.

OpenStudy (jamesj):

If 5^x = x^5 then 5^(1/5) = x^(1/x) Consider now the function f(x) = x^(1/x). As f'(x) = 1/x^2 ( 1 - ln x) . f(x) = 0 => ln x = 1 this function f has a local extrema at x = e; and it turns out it's a maximum of e^(1/e). On the interval (0,e), f(x) is monotonically increasing. And on (e,infinity) monotonically decreasing. Therefore there are indeed two solutions. We know there is one at x = 5, and the other one is a number a < e^(1/e) such that a^(1/a) = 5^(1/5). We know it exists and we know on what interval it sits. There is however no expression in elementary functions with which we can write it. If you want to estimate it, you'll need now to use some numerical method.

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