B

Step 1: Differentiate the function.
The given function is \( y = x^3 - 6x^2 + 9x + 15 \).
We differentiate this with respect to \( x \):
\( \frac{dy}{dx} = 3x^2 - 12x + 9 \).
Step 2: Set the derivative to zero to find critical points.
Solve \( 3x^2 - 12x + 9 = 0 \):
Dividing by 3:
\( x^2 - 4x + 3 = 0 \).
Factoring: \( (x - 1)(x - 3) = 0 \).
Thus, \( x = 1 \) and \( x = 3 \).
Step 3: Find the second derivative to identify maxima and minima.
\( \frac{d^2y}{dx^2} = 6x - 12 \).
Step 4: Evaluate at critical points:
For \( x = 1 \):
\( \frac{d^2y}{dx^2} = 6(1) - 12 = -6 \), indicating a maximum.
For \( x = 3 \):
\( \frac{d^2y}{dx^2} = 6(3) - 12 = 6 \), indicating a minimum.
Step 5: Calculate the values of \( y \) at critical points:
At \( x = 1\):
\( y(1) = 1 - 6 + 9 + 15 = 19 \).
At \( x = 3\):
\( y(3) = 27 - 54 + 27 + 15 = 15 \).
Therefore, the maximum value is 19 and the minimum value is 15.