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# Solution for Determine Two Positive Numbers Whose Sum is 15 and the Sum of Whose Squares is Maximum. - CBSE (Science) Class 12 - Mathematics

#### Question

Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.

#### Solution

$\text { Let the two positive numbers be x and y}. Then,$

$x + y = 15 . . . \left( 1 \right)$

$\text{Now},$

$z = x^2 + y^2$

$\Rightarrow z = x^2 + \left( 15 - x \right)^2 \left[ \text { From eq } . \left( 1 \right) \right]$

$\Rightarrow z = x^2 + x^2 + 225 - 30x$

$\Rightarrow z = 2 x^2 + 225 - 30x$

$\Rightarrow \frac{dz}{dx} = 4x - 30$

$\text { For maximum or minimum values of z, we must have }$

$\frac{dz}{dx} = 0$

$\Rightarrow 4x - 30 = 0$

$\Rightarrow x = \frac{15}{2}$

$\frac{d^2 z}{d x^2} = 4 > 0$

$\text { Substituting x } = \frac{15}{2} in \left( 1 \right), \text { we get }$

$y = \frac{15}{2}$

$\text { Thus, z is minimum when x = y } = \frac{15}{2} .$

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Solution Determine Two Positive Numbers Whose Sum is 15 and the Sum of Whose Squares is Maximum. Concept: Graph of Maxima and Minima.
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