Intersection of two lines and the minimum of the sum of the two.

Question

We use a formula in my Operations Research class for finding the 'Economic Order Quantity', given the cost function (sum of Holding and Ordering costs)

$$C = \frac{Q}{2}H+\frac{D}{Q}S$$ where $Q$ is the order size and $D$, $S$ and $H$ are constants...

When you differentiate with respect to $Q$, set equal to zero and solve for $Q$ to find the value that minimizes cost you get: $$Q=\sqrt{\frac{2DS}{H}} $$

This is also the value of $Q$ where the two lines for Holding Cost and Ordering Cost intersect. My question is why is this the case? Clearly the minimum of the sum of two lines is not always at the intersection, but I cannot see why it must be the case for this example.

Thanks for your help!

Edit:
I think my question was a little confusing. What I want to understand is: for what types of lines does the property "the minimum of the sum of the two lies at the intersection of the two" hold. (E.g., this is not true for the lines $y=x^2$ and $y=x+1$, where the minimum of the sum is at $x=-0.5$)

Answer 1

You have two functions, say $f$ and $g$. They intersect when $f(x)=g(x)$. On the other hand, their sum is minimized when $f'(x)+g'(x)=0$. In general, there is no relationship between these two conditions.

In your particular case, though, you have $f(x)$ proportional to $x$ and $g(x)$ proportional to $1/x$, so let's see what happens if we restrict our attention to power laws. Take $f(x)=ax^m$ and $g(x)=bx^n$. Then the intersection occurs at $x=(b/a)^{1/(m-n)}$, while the extremum occurs at $x=(-nb/ma)^{1/(m-n)}$. These are equal when $m=-n$, that is, when the exponents in the two functions are equal and opposite, like $1$ and $-1$ in your example.

Answer 2

The derivative of holding costs : $$\frac{1}{2}H$$ equals the negation of the ordering costs at that point

$$-\frac{D}{Q^2}S=-\frac{D}{\sqrt{\frac{2DS}{H}}^2}S=-\frac{D}{\frac{2DS}{H}}S=-\frac{H}{2}$$

Thus the derivative of the overall costs is zero at that point which indicates the presence of an extreme.

Edit

Generally: if the derivative of one function equals the negation of derivation of the the second function at the point of intersection, then the sum has derivation zero and the intersection is a (possible) extreme.