564. $y=x^{2}+2$.

Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.

564. $y=x^{2}+2$.
Solution. $1^{0}$. Find the derivative: $y^{\prime}=\left(x^{2}+2\right)^{\prime}=2 x$.

$2^{0}$. Set it to zero; $2 x=0$, from which $x=0$ - the critical point.

$3^{\circ}$. Determine the sign of the derivative at the value $x0$, for example at $x=1$: $y_{x=1}^{\prime}=2 \cdot 1=2$. Since the derivative changes sign from negative to positive as we pass through $x=0$, the function has a minimum at $x=0$.

$4^{0}$. Find the minimum value of the function, i.e., $f(0)=0^{2}+2=2$.

Now we can sketch the curve near the point $A(0 ; 2)$ (Fig. 121).