The problem in the image asks to find the minimum value of the quadratic function $f(x) = 2x^2 + 24x + 74.4$.

To find the minimum value of a quadratic function in the form $f(x) = ax^2 + bx + c$, we can follow these steps:

1. Identify coefficients

In this case, the quadratic function is: $f(x) = 2x^2 + 24x + 74.4$

• $a = 2$
• $b = 24$
• $c = 74.4$

2. Find the vertex

The x-coordinate of the vertex for a quadratic function is given by the formula: $x_{\text{vertex}} = -\frac{b}{2a}$ Substitute the values for $a$ and $b$: $x_{\text{vertex}} = -\frac{24}{2(2)} = -\frac{24}{4} = -6$

3. Calculate the minimum value

To find the minimum value, substitute $x = -6$ into the original function $f(x)$: $f(-6) = 2(-6)^2 + 24(-6) + 74.4$ $f(-6) = 2(36) + (-144) + 74.4$ $f(-6) = 72 - 144 + 74.4$ $f(-6) = 2.4$

Final Answer:

The minimum value of the function is 2.40.

Do you want more details or have any questions?

Here are some related questions:

1. How do you find the maximum value of a quadratic function?
2. What role does the coefficient $a$ play in determining whether the function has a minimum or maximum value?
3. Can you explain how the discriminant relates to the number of real roots in quadratic equations?
4. How do we derive the formula for the vertex of a parabola?
5. How do the transformations of the function $f(x) = ax^2 + bx + c$ affect its graph?

Tip: The sign of $a$ (positive or negative) tells you whether the parabola opens upwards (minimum) or downwards (maximum).