Divide 10 into two parts such that the sum of double of the first and the square of the second is minimum

Options

A. ` ( 6, 4 ) `
B. ` ( 7, 3 ) `
C. ` (8, 2 ) `
D. ` ( 9, 1 ) `

Answer

D

Explanation

Let x and y be the two parts of the number 10.
` therefore x + y = 10 " " `… (i)
According to the question,
Let ` A = 2x + y^ 2 `
= `2x + ( 10 - x )^ 2 `
` = 2x + 100 + x^ 2 - 20 x `
` = x^ 2 - 18 x + 100 `
On differentiating w.r.t. x, we get
` (dA)/ (dx ) = 2x - 18 `
For max or min of A,
Put ` (dA ) /( dx ) =0 = 2x- 18 `
` rArr x = 9 `
Now, ` (d^ 2A ) / (dx^ 2 ) = 2 gt 0 " " ` (min )
On putting ` x = 9 ` in Eq. (i), we get
` y = 1 `
` therefore ( x , y ) = ( 9, 1 ) `