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Let's break down and solve the given problem. The function presented is:

\[
f(x) = -8x^2 - 40x + 5
\]

The tasks are:

### (a) Find the vertex
The vertex of a quadratic function in the form \(f(x) = ax^2 + bx + c\) can be found using the formula for the x-coordinate of the vertex:

\[
x = -\frac{b}{2a}
\]

In this case, \(a = -8\) and \(b = -40\). Plugging these values into the formula:

\[
x = -\frac{-40}{2(-8)} = \frac{40}{-16} = -2.5
\]

Now, to find the y-coordinate of the vertex, substitute \(x = -2.5\) into the original equation:

\[
f(-2.5) = -8(-2.5)^2 - 40(-2.5) + 5
\]

\[
f(-2.5) = -8(6.25) + 100 + 5 = -50 + 100 + 5 = 55
\]

Thus, the vertex is \((-2.5, 55)\).

### (b) Maximum or minimum value
Since the coefficient of \(x^2\) (i.e., \(a = -8\)) is negative, the parabola opens downward. This means the function has a **maximum value** at the vertex. The maximum value is \(55\), which occurs at \(x = -2.5\).

### (c) Find the range
Since the parabola opens downward and the maximum value of the function is \(55\), the range of the function is:

\[
(-\infty, 55]
\]

### (d) Find the intervals of increase and decrease
- The function is **increasing** before the vertex. So, it is increasing on the interval \((-\infty, -2.5)\).
- The function is **decreasing** after the vertex. So, it is decreasing on the interval \((-2.5, \infty)\).

### Final answers:
(a) The vertex is \((-2.5, 55)\).

(b) The function has a **maximum value** of \(55\).

(c) The range of the function is \((- \infty, 55]\).

(d) The function is increasing on \((-\infty, -2.5)\) and decreasing on \((-2.5, \infty)\).

---

Would you like any more details on any of these steps?

Here are some related questions you might find useful:
1. How do you derive the vertex formula for a quadratic equation?
2. Why does the sign of the \(a\)-coefficient determine whether a parabola opens upwards or downwards?
3. How can you verify the maximum or minimum value using derivatives?
4. Can you always use the vertex form to find the range of a quadratic function?
5. How does the discriminant affect the shape of a parabola?

**Tip**: Always check the sign of the leading coefficient to quickly determine whether a quadratic function has a maximum or minimum.

Given the function f(x) = -8x^2 - 40x + 5, find (a) the vertex, (b) whether there is a maximum or minimum value, and find that value, (c) the range, and (d) the intervals on which the function is increasing or decreasing.

Learn how to find the vertex, maximum value, range, and intervals of increase and decrease for the quadratic function f(x) = -8x^2 - 40x + 5. This detailed step-by-step solution explores key algebraic concepts related to quadratic functions, suitable for high school students.

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