Mathematics 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.

8 years ago OpenStudy (jamesj):

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

8 years ago OpenStudy (jamesj):

There other one is more subtle.

8 years ago OpenStudy (anonymous):

5=x is that right

8 years ago OpenStudy (anonymous):

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

8 years ago 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

8 years ago OpenStudy (anonymous):

okay, and the other point of intersection?

8 years ago OpenStudy (jamesj):

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

8 years ago OpenStudy (anonymous):

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

8 years ago OpenStudy (anonymous):

I still need to know this.

8 years ago 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.

8 years ago
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