 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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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 $$\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}={4}^{{2}}+{4}^{{2}}+{4}^{{2}}$$
$$\displaystyle={16}+{16}+{16}={48}$$