Let's analyze the function $f(x) = 3x^2 - 24x - 4$, as requested in the image.

Part a: Determine if the function has a minimum or maximum value.

This is a quadratic function of the form $f(x) = ax^2 + bx + c$, where $a = 3$, $b = -24$, and $c = -4$.

For a quadratic function:

• 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 has a minimum value.

Part b: Find the minimum value and determine where it occurs.

To find the minimum value, we first need to find the vertex of the parabola, which occurs at $x = \frac{-b}{2a}$.

$x = \frac{-(-24)}{2(3)} = \frac{24}{6} = 4$

Now, substitute $x = 4$ back into the function to find the minimum value:

$f(4) = 3(4)^2 - 24(4) - 4 = 3(16) - 96 - 4 = 48 - 96 - 4 = -52$

Thus, the minimum value is $-52$, and it occurs at $x = 4$.

Part c: Identify the function's domain and range.

• The domain of a quadratic function is all real numbers, i.e., $(-\infty, \infty)$.
• The range depends on the vertex and the direction of the parabola:
  • Since the parabola opens upwards (because $a > 0$), the range starts from the minimum value $-52$ and goes to infinity.
  • Therefore, the range is $[-52, \infty)$.

Summary:

• a. The function has a minimum value.
• b. The minimum value is $-52$ at $x = 4$.
• c. Domain: $(-\infty, \infty)$, Range: $[-52, \infty)$.

Would you like further details or explanations?

Here are 5 related questions you might find useful:

1. How do you determine the vertex of any quadratic function?
2. What happens when the leading coefficient $a$ is negative in a quadratic function?
3. How do you find the axis of symmetry for a quadratic equation?
4. Can quadratic functions have both a minimum and maximum value?
5. How do transformations affect the graph of a quadratic function?

Tip: Remember, the sign of the leading coefficient $a$ determines whether a quadratic function has a minimum or maximum!