Find the values of $c_1,c_2$ so that $(-0.5, 0, 0)$ is a point of a local maximum.

Question

Given the problem $$ \max [-(x_1+x_2+x_3)]$$ subject to the contraints $$x_1^2+x_2^2=2c_1$$ $$x_1+5x_2+x_3^2=2c_2$$ I am asked to find the values of $c_1,c_2$ so that $(-0.5, 0, 0)$ is a point of a local maximum. I have found that $c_1=\frac{1}{8}$ and $c_2=-\frac{1}{4}$. Is this correct?

Answer 1

I was all set to build the Lagrangian and look at the optimality conditions, but it seems like there's more information here than needed. After all, for $(-0.5,0,0)$ to satisfy the constraints, you need $c_1=1/8$ and $c_2=-1/4$. No other values of $c_1$ and $c_2$ are even feasible. That is not to say that $(-0.5,0,0)$ is a local maximum, however.