If the function $f(x) = ax^3 + bx^2 + 11x - 6$, defined on $[1, 3]$, satisfies all the conditions of Rolle's theorem for $c = 2 + \dfrac{1}{\sqrt{3}}$, then

The tangent to the curve $y^2 - xy + 9 = 0$ is vertical when

The value of $k$ such that the function $f(x) = \sin x - \cos x - kx + b$ is strictly decreasing for all real $x$ is

$\int \dfrac{\sin x}{\sin 4x} dx = $