<s>[INST] Analytically minimize $x^2 + y^2$ constrained to $(x+1)^3 =- y^2$ I have the following problem:
$$\text{min} \ x^2 + y^2$$
$$s.t. \ (x+1)^3 =- y^2$$
What I did was substituting, so I got the function $f(x) = x^2  - (x+1)^3 $ but I don't know how to get analytically to the minimum. Graphing the original problem gives $x = -1$ and $y=0$ as the solution but I can't get there using my new function, I did the first derivative equal to $0$ but that has no real solutions. I did Lagrange method on the original problem but it was nonsense too. Thanks for the help.
 [/INST] A Pre-Calculus Approach
Since $(x+1)^3=-y^2$, we know that $x\le-1$. Furthermore,
$$
\begin{align}
x^2+y^2
&=\overbrace{1-2(x+1)+(x+1)^2}^{x^2=(1-(x+1))^2}-(x+1)^3\tag1\\
&=1+2u+u^2+u^3\tag2\\
&\ge1\tag3
\end{align}
$$
Explanation:
$(1)$: $y^2=-(x+1)^3$
$(2)$: substitute $u=-(x+1)$
$(3)$: $u\ge0$ since $x\le-1$
Since $(x,y)=(-1,0)$ satisfies $(x+1)^3=-y^2$ and gives $x^2+y^2=1$, we see that $1$ is the minimum of $x^2+y^2$.


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