Question

Consider an equation `log_2 (alpha^2-16alpha^3 +66) + sqrt(4beta^4 -8beta^2 +13)+ | (gamma/3-2)| = 4`, Find the numbers of ordered triplets `(alpha, beta, gamma)`? Also find the sum of all possible values of the product `alpha beta gamma`?

Answer 1

There are two solutions:

`(alpha, beta, gamma) = (2, +-1, 6)`

Hence the sum of possible values of `alphabetagamma` is `0`.

Explanation:

The question should have had `alpha^6` instead of `alpha^2` (checked against original question sheet).

Given:

`log_2(alpha^6-16alpha^3+66)+sqrt(4beta^4-8beta^2+13)+abs(gamma/3-2) = 4`

Let us look at each subexpression in turn:

`color(white)()`
(`bb alpha`):

`alpha^6-16alpha^3+66 = (alpha^3)^2-16(alpha^3)+64+2`

`color(white)(alpha^6-16alpha^3+66) = (alpha^3-8)^2+2`

`color(white)(alpha^6-16alpha^3+66) >= 2`

taking the minimum value `2` only when `alpha^3=8`, that is when `alpha=2`.

So:

`log_2(alpha^6-16alpha^3+66) >= log_2 2 = 1`

only taking the minimum value `1` when `alpha=2`.

`color(white)()`
(`bb beta`):

`4beta^2-8beta^2+13 = 4beta^2-8beta^2+4+9`

`color(white)(4beta^2-8beta^2+13) = 4((beta^2)^2-2beta^2+1)+9`

`color(white)(4beta^2-8beta^2+13) = 4(beta^2-1)^2+9`

taking the minimum value `9` when `beta^2 = 1`, i.e. when `beta = +-1`.

Hence:

`sqrt(4beta^2-8beta^2+13) >= sqrt(9) = 3`

taking the minimum value `3` when `beta = +-1`.

`color(white)()`
(`bb gamma`):

`abs(gamma/3-2)`

takes its minimum possible value `0` when:

`gamma/3-2 = 0`

That is, when `gamma = 6`

`color(white)()`
Sum:

So the minimum possible value of:

`log_2(alpha^6-16alpha^3+66)+sqrt(4beta^4-8beta^2+13)+abs(gamma/3-2)`

is `1+3+0 = 4`, only occuring when `alpha=2`, `beta=+-1` and `gamma=6`.

So the only possible solutions of the original equation are:

`(alpha, beta, gamma) = (2, +-1, 6)`

Hence, the sum of all possible values of `alphabetagamma` is:

`(2*1*6)+(2*(-1)*6) = 12-12=0`

Answer 2

See below.

Explanation:

We have a relationship as

`f(alpha)+g(beta)+p(gamma) = 4`

with

%%%%%%%%%%%%%%%%%%%
`f(alpha)=log_2(alpha^2 - 16 alpha^3 + 66)`
`g(beta) = sqrt(4 beta^4 - 8 beta^2 + 13)`
`p(gamma) = abs(gamma/3-2)`
%%%%%%%%%%%%%%%%%%%

Considering

`f(alpha)=log_2(alpha^2 - 16 alpha^3 + 66)`

the conditions on `f(alpha)` are:

`alpha^2 - 16 alpha^3 + 66 > 0`
`0 le f(alpha) le 4`

or

`1.48314 le alpha < 1.62487`

The conditions for `g(beta)` are:

`4 b^4 - 8 b^2 + 13 ge 0`
`0 le g(beta) le 4`

for

`-sqrt[1/2 (2 + sqrt[7])] le beta le sqrt[1/2 (2 + sqrt[7])]`

The conditions for `p(gamma)` are:

`0 le p(gamma) le 4`

giving

`-6 le gamma le 18`

Attached a plot showing the variety

`f(alpha)+g(beta)+p(gamma) = 4`