611. Find the number which, when added to its square, gives the smallest sum.
Solution. Let the required number be $x$. Then we need to find such an $x$ for which the function $y=x+x^{2}$ has a minimum. We find this value of $x$:

$$
y^{\prime}=1+2 x ; 1+2 x=0 ; 2 x=-1 ; x=-1 / 2 ; y^{\prime \prime}=2, y^{\prime \prime}(-1 / 2)=2>0
$$

i.e., the minimum is achieved at $x=-1 / 2$. Therefore, the required number is $-1 / 2$.