Find the minima and maxima of the function $f(x,y) = x^2 + y^2$ under the constraint $y = x^2 - 9/2$. Use Lagrange multiplier method.
So we have the function $f(x,y) = x^2 + y^2$ and I rewrite the constraint as $g(x,y) = x^2 - y - 9/2$. Then, I find the gradient vectors to be $\nabla f = (2x, 2y)$ and $\nabla g = (2y, -1)$. This gives me the system of equations
$$\begin{align} 2x &= \lambda 2x \\ 2y &= -\lambda \\ y &= x^2 - 9/2 \end{align}$$
Solving the system of equations, I get $x \pm 4$. Plugging into $y = x^2 - 9/2$, I get the critical points $(4, 23/2)$ and $(-4, 23/2)$ or $(8, 23)$ and $(-8, 23)$.
So assuming I have done everything right, I have two critical points, but I am not sure where to go from here because I am confused how I am supposed to interpret this geometrically in my head what is going on.
We have \begin{eqnarray*} L= x^2+y^2+ \lambda(y-x^2+9/2) \end{eqnarray*} This gives \begin{eqnarray*} 2x &= \lambda 2x \\ 2y &= -\lambda \\ y &= x^2 - 9/2 \end{eqnarray*} Solving these gives $\lambda=1$, $y=-1/2$, $x= 2$. Which looks like a plausible minimum.