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Question
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
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Solution
\[\text { Let the two positive numbers be x and y}. \text{ Then, }\]
\[x + y = 15 ........ \left( 1 \right)\]
\[\text{Now}, \]
\[z = x^2 + y^2 \]
\[ \Rightarrow z = x^2 + \left( 15 - x \right)^2 ..........\left[ \text { From eq } . \left( 1 \right) \right]\]
\[ \Rightarrow z = x^2 + x^2 + 225 - 30x\]
\[ \Rightarrow z = 2 x^2 + 225 - 30x\]
\[ \Rightarrow \frac{dz}{dx} = 4x - 30\]
\[\text { For maximum or minimum values of z, we must have }\]
\[\frac{dz}{dx} = 0\]
\[ \Rightarrow 4x - 30 = 0\]
\[ \Rightarrow x = \frac{15}{2}\]
\[\frac{d^2 z}{d x^2} = 4 > 0\]
\[\text { Substituting x } = \frac{15}{2} \text{ in }\left( 1 \right), \text { we get } \]
\[y = \frac{15}{2}\]
\[\text { Thus, z is minimum when x = y } = \frac{15}{2} .\]
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