Find out the answer to "Find three positive numbers whose sum is 12..."

College
Calculus and Analysis
Differential Calculus
Derivatives

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Answered question

2021-11-24

Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.

Answer & Explanation

Walker Funk

Beginner2021-11-25Added 13 answers

$F (x, y, z) = x^{2} + y^{2} + z^{2}, g (x, y, z) - x + y + z = 12$
On finding the corresponding partial derivatives of F and g and equating,
$2 x = λ$
$2 y = λ$
$2 z = λ$
$x = y = z = \frac{λ}{2}$
$x + y + z = 12 \to 3 x = 12 \to x = 4$
$F (x, y, z)$ has a minimum at x=y=z=4
Minimum value $x^{2} + y^{2} + z^{2} = 4^{2} + 4^{2} + 4^{2}$ $= 16 + 16 + 16 = 48$

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