The correct option is D (−∞,0)
Let f(x)=x2−(k+2)x+7k4
Case 1: Both the roots are negative
(i) D≥0
D=(k+2)2−7k≥0
⇒k2−3k+4≥0⇒(k−32)2+74≥0⇒k∈R
(ii) ca>0⇒k>0
(iii) −ba<0⇒k+2<0⇒k<−2
∴k∈ϕ ⋯(1)
Case 2: One root is positive and other root is negative
⇒ ca<0
⇒k<0 ⋯(2)
Checking boundary condition for case :2
For k=0, we get
x2−2x=0
⇒x=0,2
Here, no root is negative.
∴k∈(1)∪(2)
⇒k∈ϕ∪(−∞,0)
⇒k∈(−∞,0)