For n = 2 use the discriminant <0.
Maybe this idea is a little simplistic for these problem
obviously n has to be even, because i need the signs of the limits of the quadratic to be the same, when x tend to both +ve and -ve infinity. Then the graph of f(x) will curve either upwards or downwards at both ends, so there will exists some k where the equation has no solution. But how to give a formal solution??
or is there any other way to solve it???