simply simple

29/7>k>3

arey bhai ...solution do ....answer to mere paas bhi hai

btw ..your answer does not match with that given in the book
the answer given is k<-3 or 3<k≤ 29/7

x² -2(7-k)-(9-k²) = f'(x)

for 1 or more soln.

make D >=0

we get 29/7>=k ............

now to make sure its maxima .... f''(x)<0 ... which it is at the first soln as its a concave upward parabola [f'(x)]......

now we need to make sure first soln is after 0 ........

so at x=0 f'(x) > 0 ......

we get .. either k>3 or k<-3 .............

and f''(0) < 0

or 7>k ,,,,,

but 29/9 is not included else the point where f'(x) =0 .. f''(x) also will be zero :((

takin intersection of all solutions

29/7>k>3

or k<-3

sorry ..i didn't understand
btw thank you for the solution

hehe , anywyas good solution shraman.