610. Decompose the number 100 into two addends so that their product is the largest possible.
Solution. Let the first number be denoted by $x$. Then the second number is $100-x$. We then form the function $y=x(100-x)$. According to the condition, the argument $x$ must be such that the function takes its maximum value, i.e., the product on the right side of the equation is the largest. Therefore, we will find the value of $x$ at which the function has a maximum:

$$
y=100 x-x^{2} ; \quad y^{\prime}=100-2 x ; \quad 100-2 x=0 ; \quad x=50
$$

Find the second derivative: $y^{\prime \prime}=-2$. Since $y^{\prime \prime}(50)=-2<0$, the function reaches its maximum at $x=50$. Therefore, the first number $x=50$ and the second $100-x=100-50=50$. Thus, both numbers should be the same.