Question

Divide the number 100 into two parts so that the sum of their squares is minimum.

Answer 1

Let first part be x

∴ 2^nd part = 100 – x

Sum of their squares = x² + (100 – x)²

Let f(x) = x² + (100 – x)

= x² + 10000 – 200x + x²

= 2x² – 200x + 10000

∴ f′(x) = 4x – 200

∴ f″(x) = 4

For extreme values of f(x), put f’(x) = 0

∴ 4x – 200 = 0

∴ x = 50

Also f″(x) = 4 > 0

`\implies` f(x) is minimum when x = 50

∴ The two parts be 50 and 50