Find three positive numbers whose sum is 12 and the sum of whose squares is as s

adOrmaPem6r 2021-11-24 Answered
Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.

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Walker Funk
Answered 2021-11-25 Author has 9352 answers
Lagrange multiplers Method:
\(\displaystyle{F}{\left({x},{y},{z}\right)}={x}^{{2}}+{y}^{{2}}+{z}^{{2}},{g{{\left({x},{y},{z}\right)}}}-{x}+{y}+{z}={12}\)
On finding the corresponding partial derivatives of F and g and equating,
\(\displaystyle{2}{x}=\lambda\)
\(\displaystyle{2}{y}=\lambda\)
\(\displaystyle{2}{z}=\lambda\)
Therefore, \(\displaystyle{x}={y}={z}={\frac{{\lambda}}{{{2}}}}\)
Hence, \(\displaystyle{x}+{y}+{z}={12}\to{3}{x}={12}\to{x}={4}\)
Hence, \(\displaystyle{F}{\left({x},{y},{z}\right)}\) has a minimum at x=y=z=4
Minimum value is \(\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}={4}^{{2}}+{4}^{{2}}+{4}^{{2}}\)
\(\displaystyle={16}+{16}+{16}={48}\)
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