Example 6. Given a positive number a, decompose it into two addends so that their product is the largest possible.
Solution. Let $x$ be the first addend, then the second is $a-x$. Their product represents a function of $x$:

$$
f(x)=x(a-x)=a x-x^{2}
$$

We will investigate this function. Let's find the derivatives:

$$
f^{\prime}(x)=a-2 x, f^{\prime \prime}(x)=-2
$$

The first derivative is zero when $a-2 x=0$ or $x=\frac{a}{2}$. Since the second derivative is negative, the function reaches its maximum value at $x=\frac{a}{2}$. The second addend is also equal to $\frac{a}{2}$.