Question

Divide 15 into two parts such that the sum of their squares is minimum

Answer 1

7 1/2 because 7 1/2 is multiplied by 2 we get 15.so the answer is 7 1/2

Answer 2

Let one part be x. Then, other part is 15-x
$let \: \: y = {x}^{2} + {(15 - x)}^{2} \\ y = 2 {x}^{2} - 30x + 225 \\ thus \: \: y \: \: should \: \: be \: \: minimum \\ \frac{dy}{dx} = 4x - 30 \\ \\ \frac{dy}{dx} = 0 \: \: gives \: \: 4x - 30 = 0 \\ x = \frac{30}{4} = \frac{15}{2} \\ \\ \frac{ {d}^{2}y }{d {x}^{2} } = 4 > 0. \: \: \\ so \: \: x = \frac{15}{2} \: is \: minimum \: point$
Thus the required parts are 15/2 and 15/2