## Task A-3.1.

Determine the non-negative real number $a$ such that the value of the expression

$$
a^{3}-a^{2}-2 \sqrt{a}
$$

is as small as possible.
## First Solution.

Notice that for $a \geqslant 0$ the given expression is defined.

For $a=1$, the value of the expression is -2. We will prove that this is the minimum possible value.

We have

$$
\begin{aligned}
a^{3}-a^{2}-2 \sqrt{a} & =(a-2 \sqrt{a}+1)+a^{3}-a^{2}-a-1 \\
& =(\sqrt{a}-1)^{2}+a^{2}(a-1)-(a-1)-2 \\
& =(\sqrt{a}-1)^{2}+\left(a^{2}-1\right)(a-1)-2 \\
& =(\sqrt{a}-1)^{2}+(a-1)^{2}(a+1)-2
\end{aligned}
$$

Squares of real numbers are always non-negative, and the expression $a+1$ is positive since $a \geqslant 0$.

Therefore, $a^{3}-a^{2}-2 \sqrt{a} \geqslant-2$ for all $a \geqslant 0$.