Using a graphical method, indicate the feasible region and solve the minimization problem. $$\begin{array}{ll} \text{minimize} & f := x_1^2 + x_2 + 4\\ \text{subject to} & c_1 := -x_1^2-(x_2+4)^2 +16 \ge 0\\ & c_2 := x_1 - x_2 - 6 \ge 0\end{array}$$
I draw the problem as such:
And I do understand that to be able to relate the constraints to the subject function, the function needs to be held constant. I do have a hard time understanding where the minimizer actually is.
By holding the function constant results in level curves consisting of parabolas, is the minimizer at the top of those parabolas, or have i misunderstood something?
First, a small mistake in your figure, the line for the second constraint does not pass through $(0,-4)$.
The function that you want to minimize is $f(x)=x_1^2+x_2+4$. Let's say the value for this is some $v$, not necessarily the minimum. We want to plot contours where $f(x)=v$. $$x_2=-x_1^2-4+v$$ These are upside down parabolas, shifted along $x_2$ axis, with the vertex on the $x_1=0$. The the figure below. The legend shows the $v$ value.
It is easy to see that the minimum of $f(x)$ given the constraints is $-4$ at $x=(0,-8)$.