For any value of x the expression `2(k-x)(x+sqrt(x^2+k^2))` cannot exceed

Options

A. `k^(2)`
B. `2k^(2)`
C. `3k^(2)`
D. none of these

Answer

B

Explanation

Let `y=2(kpx)(x+sqrt((x^(2)+k^(2)))`
`impliesy-2(k-x)x=2(k-x)sqrt((x^(2)+k^(2)))`
On squaring both sides we get
`impliesy^(2)+4(-x)^(2)x^(2)-4xy(k-x)=4(k-x)^(2)(x^(2)+k^(2))`
`impliesy^(2)-4xy(k-x)=4(k-x)^(2)k^(2)`
`implies4(k^(2)-y)x^(2)-4(2k^(3)-ky)x-y^(2)+4k^(4)=0`
Since `x` is real
`:.Dge0`
`implies16(2k^(3)-ky)^(2)-4.4(k^(2)-y)(4k^(4)-ky^(2))ge0`
[using `b^(2)-4acge0`]

`implies4k^(6)+k^(2)y^(2)-4k^(4)y-(-k^(2)y^(2)+4k^(6)+y^(3)-4yk^(4))ge0`
`implies2k^(2)y^(2)-y^(3)ge0`
`impliesy^(2)(y-2k^(2))le0`
`:.yle2k^(2)`