Learn / Library / Calculus

Balancing delivery costs

CalculusMedium

A shop's daily cost, in hundreds of dollars, is $C (x) = x^{2} + \frac{16}{x}$ when it orders $x$ crates per delivery, where $x$ can be any real number with $x > 0$ . What is the smallest possible daily cost, in hundreds of dollars? Give the answer as an integer.

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Set the derivative to zero and check that the point is a minimum.

Solution

$C^{'} (x) = 2 x - \frac{16}{x ^{2}}$ , which is zero when $x^{3} = 8$ .
So the only critical point is $x = 2$ ; $C^{'}$ is negative before it and positive after it, so it is the minimum.
The minimum cost is $C (2) = 4 + \frac{16}{2} = 12$ .

Answer: $12$

Next question Back to the library