5 Given the following equation $f(x)=A x^{3}+B x^{2}+C x+D$. Determine $A, B, C$, and $D$ if it is known that the graph of this function has an inflection point at $(1,0)$ and has an extreme value at $(2,1)$.

Step 1 ：We are given the cubic function \(f(x)=A x^{3}+B x^{2}+C x+D\). We need to find the coefficients A, B, C, and D. We know that the function has an inflection point at (1,0) and an extreme value at (2,1).

Step 2 ：The inflection point of a cubic function is the point where the second derivative equals zero. So, we set the second derivative of the function equal to zero and substitute x=1 to get the equation \(6A + 2B = 0\).

Step 3 ：The extreme value of a cubic function is the point where the first derivative equals zero. So, we set the first derivative of the function equal to zero and substitute x=2 to get the equation \(12A + 4B + C = 0\).

Step 4 ：Substituting the coordinates of the inflection point (1,0) into the function gives us the equation \(A + B + C + D = 0\).

Step 5 ：Substituting the coordinates of the extreme value (2,1) into the function gives us the equation \(8A + 4B + 2C + D = 1\).

Step 6 ：Solving this system of equations gives us the solution \(A = -\frac{1}{2}\), \(B = \frac{3}{2}\), \(C = 0\), and \(D = -1\).

Step 7 ：\(\boxed{\text{Final Answer: The coefficients of the cubic function are } A = -\frac{1}{2}, B = \frac{3}{2}, C = 0, \text{ and } D = -1. \text{ So the function is } f(x) = -\frac{1}{2}x^3 + \frac{3}{2}x^2 - 1}\)