The given production cost function is:

$C(x) = 2x^2 - 40x + 300$

To find the minimum production cost, we need to determine the vertex of this quadratic function. Since the function is in the form of $C(x) = ax^2 + bx + c$, where $a = 2$, $b = -40$, and $c = 300$, the x-coordinate of the vertex (which gives the value of $x$ that minimizes the cost) can be found using the formula:

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

Let's calculate that first:

$x = -\frac{-40}{2(2)} = \frac{40}{4} = 10$

So, the minimum production cost occurs when 10 necklaces are produced daily.

Next, we find the minimum production cost by substituting $x = 10$ into the cost function:

$C(10) = 2(10)^2 - 40(10) + 300$ $C(10) = 2(100) - 400 + 300$ $C(10) = 200 - 400 + 300 = 100$

Thus, the minimum production cost is $100, and it occurs when 10 necklaces are produced daily.

Summary:

• Minimum production cost: $100
• Corresponding daily number of necklaces produced: 10

Would you like further details or have any questions?

Here are five related questions:

1. What happens to the cost if the number of necklaces produced increases or decreases from 10?
2. How does changing the coefficient of $x^2$ in the cost function affect the minimum cost?
3. What is the maximum cost for producing one necklace if the number of necklaces produced is limited to less than 10?
4. How would the function change if the fixed cost (300) increases by 50%?
5. What is the profit function if each necklace sells for $20?

Tip: For any quadratic function, the vertex formula helps find the minimum or maximum values, essential in optimization problems.