Consider the function \( f(x)=6 x^{2}-6 x \). a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. a. The function has a maximum

value because the coefficient of \( x^2 \) is positive, which indicates that the parabola opens upwards. This means that the vertex of the parabola, which is the point where the function reaches its maximum or minimum, will be the minimum point in this case. b. To find the minimum or maximum value, we need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients of \( x^2 \) and \( x \) in the quadratic equation, respectively. In this case, \( a = 6 \) and \( b = -6 \), so the x-coordinate of the vertex is \( x = -\frac{-6}{2 \cdot 6} = \frac{1}{2} \). To find the y-coordinate of the vertex, we substitute the x-coordinate back into the original function: \( f\left(\frac{1}{2}\right) = 6\left(\frac{1}{2}\right)^2 - 6\left(\frac{1}{2}\right) = 6 \cdot \frac{1}{4} - 6 \cdot \frac{1}{2} = \frac{3}{2} - 3 = -\frac{3}{2} \). Therefore, the minimum value of the function is \( -\frac{3}{2} \), and it occurs at \( x = \frac{1}{2} \). c. The domain of the function is all real numbers, since there are no restrictions on the input values. Therefore, the domain is \( (-\infty, \infty) \). The range of the function is all real numbers greater than or equal to the minimum value, which is \( -\frac{3}{2} \). Therefore, the range is \( \left[-\frac{3}{2}, \infty\right) \).