To find the minimum cost per unit, we need to analyze the cost function given:

$C(x) = 7x^2 + 252$

Here, $C(x)$ represents the total cost for ordering $x$ units, so the cost per unit would be:

$\frac{C(x)}{x} = \frac{7x^2 + 252}{x}$

Simplifying this expression:

$\frac{C(x)}{x} = 7x + \frac{252}{x}$

To minimize the cost per unit, we need to find the minimum of this function. This requires taking the derivative with respect to $x$ and setting it to zero:

1. Differentiate: $\frac{d}{dx}\left(7x + \frac{252}{x}\right) = 7 - \frac{252}{x^2}$

2. Set the derivative equal to zero to find the critical points: $7 - \frac{252}{x^2} = 0$

   Solving for $x^2$: $\frac{252}{x^2} = 7$ $x^2 = \frac{252}{7} = 36$ $x = \sqrt{36} = 6$

So, the minimum cost per unit occurs when $x = 6$.

Answer:

The correct option is: B. An order of 6 units has a minimum cost per unit.

Would you like further explanation or have any questions?

Related Questions

1. How do we confirm that this is a minimum and not a maximum?
2. What happens to the cost per unit if we increase $x$ beyond 6 units?
3. How does the fixed cost in $C(x)$ affect the cost per unit?
4. Can this technique be used for other cost functions with similar structures?
5. How do we interpret the result in terms of business efficiency?

Tip

For functions of the form $ax^2 + b$, the minimum occurs when the variable is isolated effectively through derivative analysis.