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?

2 Answers
Mar 20, 2017

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

Mar 20, 2017

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

enter image source here