What are the intervals containing the solutions for 4x^2 - e^x = 0?

[stunner5000pt]

Find the intervals containing the soltuions for $4x^2 - e^x = 0$

now this can't be solved by elementary methods since all i get is
$$x - 2 \ln(x) = \ln(4)$$
finding the derivative doesn't help either

$$f'{x} = 8x - e^x$$
$$0 = 8x - e^x$$

cant just GUESS ... that isn't right ...
this is for a numericals methods class, i do not believe that they wat me to use fixed point iteration or anything else for that matter with this problem

please help! Thank you!

 

[matt grime]

You only have to find some intervals (you can't say 'the' since there isnothing unique about them) which is simply an application of the intermediate value theorem, so find out points where it changes sign (eg x-2 has a root in the interval [1,3] because at 1 it is negative at 3 it is positive so it has a root. the harder thing is to

 

[stunner5000pt]

so all i have to do is sub in numbers to get a postiive number and a nagetive number at each end point
but what if this had many roots (say 4) within an interval such as [1,2]?

also would i be subbin in numbers such as interavls with integer end points?

 

[shmoe]

Intermediate value theorem will tell you you have a root in an interval, but doesn't eliminate the possibility of multiple roots. You need some other way of determining the number of roots (or maximum number of roots), then if you can find that many disjoint intervals that contain roots, you've located them all. You could look at first and second derivatives for example.

Integers are nicer to work with, but don't limit yourself to integer endpoints, you might not be able to find all the zeros.