Q.Find two positive numbers whose sum is $15$ and the sum of their squares is minimum.

Minimizing the Sum of Squares

A classic optimisation problem in this chapter asks: split a fixed quantity into parts so that the sum of their squares is as small as possible. For instance, "Find two positive numbers whose sum is $S$ and the sum of whose squares is a minimum." The derivative handles it neatly.

Setting it up

Suppose two numbers add up to a fixed total $S$. Call them $x$ and $S - x$. The quantity we want to minimise is the sum of their squares:

By writing the second number in terms of the first, we have turned a two-variable question into a single-variable function — exactly what makes it a maxima–minima problem.

Applying the derivative

Differentiate and set to zero:

The second derivative is $f''(x) = 4 > 0$, which is positive everywhere, so this critical point is a genuine minimum. Hence the two numbers are equal, each $\dfrac{S}{2}$, and the least possible sum of squares is

The general recipe

Whenever a problem says "minimise the sum of squares subject to a fixed sum":

1. Use the constraint to express everything in one variable. …