At how many points in the xy-plane do the graphs of x^12 and

Question

At how many points in the xy-plane do the graphs of x^12 and 2^x intersect ?


A) None
B) One
C) Two
D) Three
E) Four

Have fun,
Blueberry

christoph · Answer

C)...x^12=2x....x can be 0 and x can be a +ve value !

laxieqv · Answer

the intersection must be root of the equation: x^12= 2^x :wink:

looks like there's no root for this equation..

gamjatang · Answer

At how many points in the xy-plane do the graphs of x^12 and 2^x intersect ? 


x^12 = 2^x

If x = -2, 5096 =/= 1/4
If x = -1, 1 =/= 1/2
If x = 0, 0 =/= 1
If x = 1, 1 =/= 2
If x = 2, 5096 =/= 4

I don't think those two graphs intersect each other at any point.

laxieqv · Answer

This sort of exponential problem involves the use of logarithm to solve. This gonna be complicated ..i don't think GMAT gonna test it so the answer choice must be gained after plugging some simple numbers. It should be A.

BG · Answer

X^12 is a parabolla type of function with wings going upward and peak at the beginning of the coordinate system.
2^x is an exponential function. 
Just check X=0
think that there are three points in common or D)

christoph · Answer

confused 2x with 2^x ! when we assume that its 2x, then its C) ! :wink:

ok. y=2^x. plug in 0 for x to see where it intersects with the y-axis. its (0,1). plug in +ve values for x and u will see that the line grows very steep in II. plug in -ve value and u will see that the line grows very flat in I. y=x^12 is a parabola. plug in some values again. the 2 graphs intersect at 2 points ! so its still C) !

laxieqv · Answer

uhm, i doubt the bold part. x^2 can be a parabola ...but x^12 we are not sure even though this function has symmetric charateristics ...for example x^4 is surely not a parabola.

christoph · Answer

the bolded part is not true. y=x^12 is the equation of a parabola w/o a y-intersection ! a parabola is the set of all points in the plane which are equally distant from a fixed point !

laxieqv · Answer

https://mathworld.wolfram.com/Parabola.html

christoph · Answer

ur right ! the graph, by definiton, is not a parabola ! but the 2 graphs still intersect in 2 point ! :wink:

laxieqv · Answer

yes, i re-consider, there should be intersection points , and 2 is the most possible one :wink:

...i'm sure this one must use logarithm formulas to solve and it's not simple ...at this time, i can't recall how but i did solve this kind of problem in high school ( years ago already) ...OMG, i'm just getting older :cry:

blueberry · Answer

Guys, sorry about the confusion between 2^x and 2x. It is 2 raised to the power x, I tried using the superscript code to write it properly, but it does not work. If any of guys know how to write exponents, please let me know. 

The Answer is D. Three Points.

The graph of x^12 is symmetric about the y-axis, in technical terms it is an even function, just like x^2. At x=0 it is zero, whereas 2^x is 1. as x becomes -ve, 2^x approaches zero, therefore x^12 will intersect 2^x for some negative real value of x. Similarly, at x=2, 2^12 is greater than 2^2, therfore there is solution between, x=0 and 2. The trickiest part is to realize that as x-becomes large, then eventually 2^x will again exceed x^12. 

Consider x=2^7, then 2^x = 2^128 and x^12=(2^7)^12=2^84. 

Thanks for attempting the problem. It is a very difficult question.

blueberry

kfranson · Answer

Going with D, 3 intersection points, 2^x crosses through 2 points in the "parabolic" region of x^12, then x^12 rises faster than 2^x for a while and then as x gets larger 2^x will surpass x^12 again for the third intersection point.

laxieqv · Answer

solution like this is a little abstract ...especially you can hardly draw the graph for x^12 ...we can use logarithm to solve , i'm sure ...once i get it, i'll post here :wink:

....BTW, this one is surely not of GMAT's taste, isn't it?!!!

willget800 · Answer

you never know anymore... lots of harder questions have started coming?

any ideas on how to solve it fast, other than plotting points on the notebook?

sm176811 · Answer

<img src="{SMILIES_PATH}/icon_rolleyes.gif" alt=":roll:" title="Rolling Eyes" />

ccax · Answer

kfranson and blueberry are right. And their explanation is as well. It's
not as difficult to see as it might on the first glance. Way more difficult is
the question about the exact locations of the intersections.

This can't be solved by "normal" logarithms since you need a function
to give the solution for w in z = w * exp(w). This can be done by the
so called Lambert-W- or omega function.

The x-coordinates of the intersections are approximately
-0.946,
+1.063 and
+74.67

The linear-log plot graph is shown in the image. Pay attention to the
huge y-coordinates.

HongHu · Answer

Yes, this is a tough question isn't it. For GMAT I do not think a rigorous solution for this type of questions is required. However, one could get the correct answer by going through a more intuitive route.

We know the shape of 2^x. It approaches 0 when x -> -infinity, intercept y axis at x=0, y=1, and then increases when x->infinity. We also know that x^12 approaches infinity when x ->+/- infinity and intercept y axis at x=0, y=0.

Since in the negative x region the 2^x increases monotonically while x^12 decreases monotonically, also considering their y position at x=0, we know that there will be one intercept in this region.

It is the positive x region that are more problematic. I would have said that there was one intercept too there without more careful thinking. All we really need to know is the shape of a^x relative to x^a. Exponential curves increases at a slower speed when x is small and increases very fast (faster then polynomial curves) when x is big. An easier question of the same type would be compare 2^x and x^2. You will easily verify that there are two intercept in the positive x region (x=2 and x=4). You could also verify what their relative position is when x=1, x=3 and x=5. Knowing this general relationship between exponential curves and polynomial curves it will be a intuitive judgement to say that the same conclusion holds true for all exponential curves and polynomial curves.

At how many points in the xy-plane do the graphs of x^12 and

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