A

Step 1: Find the derivative of the function
To find the maximum and minimum values of the function
$$ y = x^3 - 6x^2 + 9x + 15 $$
we need to find its derivative:
$$ \frac{dy}{dx} = 3x^2 - 12x + 9 $$

Step 2: Set the derivative to zero
Setting the derivative to zero to find critical points:
$$ 3x^2 - 12x + 9 = 0 $$

Dividing through by 3:
$$ x^2 - 4x + 3 = 0 $$

Step 3: Factor the quadratic
Factoring gives us:
$$ (x - 1)(x - 3) = 0 $$
Thus, the critical points are:
$$ x = 1 $$ and $$ x = 3 $$

Step 4: Evaluate the function at critical points and endpoints
Now, we need to evaluate the original function at the critical points and, if necessary, at the endpoints of the domain (typically taken from negative to positive infinity for such polynomial functions). For our case, we evaluate at the critical points:
- At $$ x = 1 $$:
$$ y(1) = 1^3 - 6(1)^2 + 9(1) + 15 = 19 $$
- At $$ x = 3 $$:
$$ y(3) = 3^3 - 6(3)^2 + 9(3) + 15 = 15 $$

Step 5: Compare values
Values of the function at critical points are:
- $$ y(1) = 19 $$
- $$ y(3) = 15 $$
The maximum value is 19 at $$ x = 1 $$ and minimum value is 15 at $$ x = 3 $$.

Final Answer:
The maximum value is 19 and the minimum value is 15.