If the equation `x^(2)+2(lamda+1)x+lamda^(2)+lamda+7=0` has only negative roots, then the least value of `lamda` equals______.

Answer

6

Explanation

Let `f(x)=x^(2)+2(lamda+1)x+lamda^(2)+lamda+7` If both roots of f(x)=0 are negative, then
`D=b^(2)-4ac=4(lamda+1)^(2)-4(lamda^(2)+lamda+7)ge0`
`implieslamda-6ge0" "(1)`
`implieslamdain[6,oo)`
Sum of roots `=-2(lamda+1)lt0`
`implieslamdain(-1,oo)" "(2)`
and product iof roots `=lamda^(2)+lamda+7gt0implieslamdainR" "(3)`
`:.` From (1),(2),(3), we get `lamdain[6,oo)` (as(1),(2),(3) must be satisfied simultaneously.)
Hence, the least value of `lamda=6`.