Intersection of Polynomial and Exponential Functions

[darkchild]

Homework Statement

At how many points in the xy-plane do the graphs of y=x^{12} and y=2^{x} intersect?

Homework Equations

none

The Attempt at a Solution

I have no idea what to do. I thought of trying to narrow it down to some intervals where the graphs may cross, but, since they're both always non-negative and always increasing, I can't see how I would do that. I also thought about approximating with Newton's Method, but that could take a while. This is a practice problem from the gre math subject test, so the method to solve it should be something that is relatively quick.

 

[Gib Z]

Just draw a sketch of the two graphs. Take care to draw x^12 quite close to zero for -1 < x < 1 and draw it quite steeply increasing outside of that range (basically, don't just draw a parabola shape). Then draw 2^x. Can you see why there should be two solutions near the origin, 1 negative and another positive, and why there should be a third solution some place far away ( turns out, near x=75 ) when the exponential curve finally catches up to the polynomial, and then leaves it behind for good?

 

[I like Serena]

I've recently "discovered" Wolfram|Alpha through other posts in this forum.

Let me share its results applying to this case:

http://www.wolframalpha.com/input/?i=solve+x^12%3D2^x+for+x

As you can see it finds the 3 solutions near -1, +1, and 75.