1. The problem asks for the minimum output value of the function $$f(x) = 3(x - 1)(x - 7)$$. 2. This is a quadratic function in factored form. Since the coefficient of the quadratic term (after expansion) is positive, the parabola opens upward, so it has a minimum point. 3. To find the minimum value, first find the vertex of the parabola. The vertex's x-coordinate is the midpoint of the roots 1 and 7: $$x = \frac{1 + 7}{2} = \frac{8}{2} = 4$$ 4. Substitute $$x = 4$$ into the function to find the minimum output value: $$f(4) = 3(4 - 1)(4 - 7) = 3(3)(-3) = 3 \times 3 \times (-3) = -27$$ 5. Therefore, the minimum output value of the function is $$-27$$.