The given production cost function is:

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

This is a quadratic function in the form $C(x) = ax^2 + bx + c$, where $a = 2$, $b = -40$, and $c = 300$.

To find the minimum production cost, we need to determine the vertex of this parabola. For a quadratic function, the x-coordinate of the vertex, which gives the minimum or maximum value, is given by the formula:

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

Substituting the values of $a$ and $b$:

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

Thus, the minimum production cost occurs when 10 necklaces are produced per day.

Now, to find the corresponding minimum production cost, substitute $x = 10$ back into the cost function $C(x)$:

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

Thus, the minimum production cost is $100 when 10 necklaces are produced per day.

Summary:

• Minimum production cost = $100
• Number of necklaces produced = 10 per day.

Would you like more details on this, or do you have any questions?

Related Questions:

1. How would the production cost change if the coefficient of $x^2$ were different?
2. What happens if the production cost equation is shifted vertically?
3. How would the daily production level change if the linear term $-40x$ were modified?
4. How does the discriminant of a quadratic equation relate to production costs?
5. What is the significance of the parabola opening upwards in this context?

Tip:

Always check the sign of the coefficient of $x^2$ in a quadratic function. If it's positive, the parabola opens upwards and has a minimum point.