Question

How can we conclude that there is a number x between 2 and 3 such that q(x)=51?

Let `q(x)=x^4`. Evaluate q(2) and q(3). Explain how we can conclude that there is a number x between 2 and 3 such that q(x)=51

Answer 1

A few thoughts...

Explanation:

There are several different possible approaches to this question, but let's start with some basics:

Given:

`q(x) = x^4`

Then:

• `q(x)` is a continuous function.

• `q(2) = (color(blue)(2))^4 = 16 < 51`

• `q(3) = (color(blue)(3))^4 = 81 > 51`

Intermediate value theorem

If a function `f(x)` of real numbers is defined and continuous on an interval `[a, b]` with `f(a) < f(b)` and `y in [f(a), f(b)]` then there is some `x in [a, b]` such that `f(x) = y`.

In our example, `q(x)` is defined and continuous on `[2, 3]` (actually on the whole of `RR`), with `q(2) < q(3)` and `51 in [q(2), q(3)]`. So there is some `x in [2, 3]` such that `q(x) = 51`.

Cauchy sequence

Define a sequence recursively by:

`{ (a_1 = 2), (a_(n+1) = { (a_n - 2^(-n) " if " a_n^4 > 51), (a_n color(white)(xxxxx) " if " a_n^4 = 51), (a_n + 2^(-n) " if " a_n^4 < 51) :} ) :}`

Then for any `epsilon > 0` then is some `N in ZZ` such that for all `m, n >= N`, we have `abs(a_m - a_n) < epsilon`.

So `a_1, a_2, a_3,...` is a Cauchy sequence and hence defines a real number.

Note that:

`sum_(n=1)^oo 2^(-n) = 1`

Hence the rules we have given are sufficient to tend to any limit in `[2, 3]`.

Because of the way the recursive rule is formed, the `a_n`'s form a sequence of approximations to `root(4)(51)` with `root(4)(51)` as their limit.

Dedekind cut

Define sets `L` and `R` as follows:

`L = { x in QQ : x < 0 vv x^4 < 51 }`

`R = { x in QQ : x > 0 ^^ x^4 >= 51 }`

Then:

• `L uu R = QQ`

• `L nn R = O/`

• All of the elements of `L` are less than all of the elements of `R` and `L` contains no greatest element.

`L` and `R` form a parition of the rational numbers defining a cut point identifying the real number `root(4)(51)`

Note that `2^4 < 51` and hence `2 in L`, while `3^4 > 51` and hence `3 in R`.