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.