The cost function is given as:

$C(x) = 2x^2 - 32x + 520$

This is a quadratic function, and since the coefficient of $x^2$ is positive ($2 > 0$), the parabola opens upwards. Thus, the minimum cost occurs at the vertex of the parabola.

The formula for the x-coordinate of the vertex of a quadratic function $ax^2 + bx + c$ is:

$x = \frac{-b}{2a}$

In this case, $a = 2$ and $b = -32$. Plugging these into the formula:

$x = \frac{-(-32)}{2(2)} = \frac{32}{4} = 8$

So, the number of videos rented daily that minimizes the cost is $x = 8$.

Next, we substitute $x = 8$ back into the cost function to find the minimum cost:

$C(8) = 2(8)^2 - 32(8) + 520$ $C(8) = 2(64) - 256 + 520$ $C(8) = 128 - 256 + 520 = 392$

Therefore, the lowest cost is $392.

Would you like further details or have any questions?

Here are 5 related questions:

1. How does the vertex form of a quadratic equation relate to its graph?
2. Can the lowest cost be higher or lower if we change the coefficients of the function?
3. What happens to the vertex of a parabola if we increase or decrease the value of $a$?
4. How do you derive the formula for the vertex of a quadratic function?
5. What would the cost function look like if it represented maximum rather than minimum cost?

Tip: The vertex formula works for any quadratic function, and it's key for finding maximums and minimums efficiently!