Question

The number of integers in the range of ‘c’ such that there exists a line which intersects the curve $y=x^4–6x^3+12x^2+cx+1$ at four distinct points is equal to ___

Answer 1

$\because y=x^4-6x^3+12x^2+cx+1\Rightarrow \frac{dy}{dx}=4x^3-18x^2+24x+c=g\left(x\right)and\frac{d^{2}y}{d{x}^2}=2x^2-36x+24=12(x-1)(x-2)$
Necessary condition, g(1). g(2) < 0
$\therefore$ Number of integers = 1