In the coordinate plane, a ``Bézier curve'' determined by four points $A$, $B$, $C$, $D$ refers to a polynomial function of degree at most 3 whose graph passes through points $A$ and $D$, and whose tangent line at point $A$ passes through point $B$, and whose tangent line at point $D$ passes through point $C$. Let $y = f(x)$ be the ``Bézier curve'' determined by the four points $A(0, 0)$, $B(1, 4)$, $C(3, 2)$, $D(4, 0)$. Answer the following questions. (1) Let the equation of the tangent line to the graph of $y = f(x)$ at point $D$ be $y = ax + b$, where $a$ and $b$ are real numbers. Find the values of $a$ and $b$. (2 points) (2) Prove that the polynomial $f(x)$ is divisible by $x^{2} - 4x$. (2 points) (3) Find $f(x)$. (4 points) (4) Find the value of the definite integral $\int_{2}^{6} |8f(x)|\, dx$. (4 points)
In the coordinate plane, a ``Bézier curve'' determined by four points $A$, $B$, $C$, $D$ refers to a polynomial function of degree at most 3 whose graph passes through points $A$ and $D$, and whose tangent line at point $A$ passes through point $B$, and whose tangent line at point $D$ passes through point $C$. Let $y = f(x)$ be the ``Bézier curve'' determined by the four points $A(0, 0)$, $B(1, 4)$, $C(3, 2)$, $D(4, 0)$. Answer the following questions.\\
(1) Let the equation of the tangent line to the graph of $y = f(x)$ at point $D$ be $y = ax + b$, where $a$ and $b$ are real numbers. Find the values of $a$ and $b$. (2 points)\\
(2) Prove that the polynomial $f(x)$ is divisible by $x^{2} - 4x$. (2 points)\\
(3) Find $f(x)$. (4 points)\\
(4) Find the value of the definite integral $\int_{2}^{6} |8f(x)|\, dx$. (4 points)