\( 1 \leftarrow \) Consider the function \( f(x)=3 x^{2}-30 x-3 \) a. Determine, without graphing, whether the function has a minimum value or a maximum value b. Find the minmum or maximum value and determine where it occurs c. Identity the function's domain and its range a. The function has a minimum value b. The minimum/maximum value is \( \square \) It occurs at \( x=\square \)

The function \( f(x) = 3x^2 - 30x - 3 \) has a minimum value because the coefficient of \( x^2 \) is positive (3). A parabola that opens upwards indicates a minimum point at its vertex. To find the minimum value, we use the vertex formula \( x = -\frac{b}{2a} \), where \( a = 3 \) and \( b = -30 \). Plugging in the values: \[ x = -\frac{-30}{2 \cdot 3} = 5. \] Now, substituting \( x = 5 \) back into the function: \[ f(5) = 3(5^2) - 30(5) - 3 = 75 - 150 - 3 = -78. \] So, the minimum value is \(-78\) which occurs at \( x = 5 \). The domain of \( f(x) \) is all real numbers, \( (-\infty, \infty) \), since it's a polynomial function. The range, however, starts from the minimum value \(-78\) to positive infinity: \( [-78, \infty) \).