Find the maximum and minimum points on the graph of the function \(f(x) = x^3 - 6x^2 + 9x + 15\).

Step 1 ：The maximum and minimum points of a function occur at the points where the derivative of the function is equal to zero. So, first we find the derivative of the function \(f(x) = x^3 - 6x^2 + 9x + 15\).

Step 2 ：The derivative of \(f(x) = x^3 - 6x^2 + 9x + 15\) is \(f'(x) = 3x^2 - 12x + 9\).

Step 3 ：Setting \(f'(x) = 0\), we get \(3x^2 - 12x + 9 = 0\). Solving this quadratic equation, we get two roots: \(x = 1\) and \(x = 3\).

Step 4 ：To determine whether these points are maximum or minimum, we check the second derivative of the function at these points. The second derivative of the function is \(f''(x) = 6x - 12\).

Step 5 ：Substituting \(x = 1\) into \(f''(x)\), we get \(f''(1) = -6 < 0\). This means that \(x = 1\) is a maximum point.

Step 6 ：Substituting \(x = 3\) into \(f''(x)\), we get \(f''(3) = 6 > 0\). This means that \(x = 3\) is a minimum point.

Step 7 ：Finally, substitute \(x = 1\) and \(x = 3\) into the original function to find the y-coordinates of the maximum and minimum points: \(f(1) = 1^3 - 6*1^2 + 9*1 + 15 = 19\) and \(f(3) = 3^3 - 6*3^2 + 9*3 + 15 = 9\).