Minimum value when \(\displaystyle{a}{b}{c}+{a}{b}+{4}{b}{c}+{9}{c}{a}={144}\)

Leonard Montes

Leonard Montes

Answered question

2022-04-15

Minimum value when
abc+ab+4bc+9ca=144

Answer & Explanation

vadomosigmx6

vadomosigmx6

Beginner2022-04-16Added 12 answers

Step 1
First the constraint
144=abc+ab+4bc+9ca
is transformed into a more suitable expression:
Divide it by 36 and rescale variables by α=a4,β=b9 and γ=c. This leads to
4=αβγ+αβ+βγ+γα big+αβ+βγ+γα+4(α+β+γ)+8
[1.6ex]cyc(α+2)(β+2)=(α+2)(β+2)(γ+2) Big1RHS
cyc1α+2=1   (1)
Next apply the Cauchy–Bunyakovsky–Schwarz inequality and exploit (1) to obtain
(2+3+1)2=(2α+2·1α+2+3β+2·1β+2+γ+2·1γ+2)2
[1ex]4(α+2)+9(β+2)+γ+2
[2ex] 8 4α+9β+γ=a+b+c
Finally, one has equality only if one argument vector is a scalar multiple of the other. Thus,
2α+2=λα+2
and so on, or 2(α+2)=λ=3(β+2)=γ+2
which yields λ=6 using (1). Hence (a,b,c)=(4,0,4) is the unique minimising solution

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