Questions: Consider the function f(x)=3x^2-24x-7. 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.

Solution Steps

The function given is a quadratic function of the form \( f(x) = ax^2 + bx + c \), where \( a = 3 \), \( b = -24 \), and \( c = -7 \).

For a quadratic function \( ax^2 + bx + c \):

• If \( a > 0 \), the parabola opens upwards, and the function has a minimum value.
• If \( a < 0 \), the parabola opens downwards, and the function has a maximum value.

Since \( a = 3 > 0 \), the function \( f(x) = 3x^2 - 24x - 7 \) has a minimum value.

The minimum value of a quadratic function \( ax^2 + bx + c \) occurs at the vertex, which is given by the formula: \[ x = -\frac{b}{2a} \]

Substituting the values of \( a \) and \( b \): \[ x = -\frac{-24}{2 \times 3} = \frac{24}{6} = 4 \]

To find the minimum value, substitute \( x = 4 \) back into the function: \[ f(4) = 3(4)^2 - 24(4) - 7 \] \[ f(4) = 3(16) - 96 - 7 \] \[ f(4) = 48 - 96 - 7 \] \[ f(4) = -55 \]

Thus, the minimum value is \(-55\) and it occurs at \( x = 4 \).

The domain of any quadratic function is all real numbers, i.e., \( (-\infty, \infty) \).

Since the function has a minimum value at \( x = 4 \) and the parabola opens upwards, the range of the function is: \[ [f(4), \infty) = [-55, \infty) \]

Final Answer