For real solution of equation `3sqrt(x+3p+1)-3sqrt(x)=1`, we have

Options

A. `p ge 1//4`
B. `p ge -1//4`
C. `p ge 1//3`
D. `p ge -1//3`

Answer

B

Explanation

`(b)` `3sqrt(x+3p+1)=3sqrt(x)+1`
Let `3sqrt(x)=h`
`implies3sqrt(x+3p+1)=h+1`
`impliesx+3p+1=h^(3)+3h^(2)+3h+1`
`implies h^(3)+3p+1=h^(3)+3h^(2)+3h+1`
`implies3h^(2)+3h-3p=0`
`h^(2)+h-p=0`
For real roots ` D ge 0`
`implies 1+4p ge 0`
`implies p ge -1//4`