2016/009) What is the value of c such that a straight line exists which intersects $f(x)=x^4+9x^3+cx2+9x+4$ at 4 points

$f(x)=x^4+9x^3+cx^2+9x+4$
for a straight line to intersect at 4 distinct points
$f^{''}(x)$ must have 2 roots
The reason
if it has no root $f'(x)$ is either positive or -ve and so f(x) is monotonically
increasing or decreasing. as it is cubic polynomial with leading coefficient positive
it is increasing. hence no line can intersect at more than 2 points
if it has one zero then it does not have any point of inflection
now $f^{''}(x)=12x^2+54x+2c=12(x+\frac{9}{4})^2+2(c-\frac{243}{8})$
it has a double root when $c<\frac{243}{8}$
hence $c<\frac{243}{8}$