I'm asked to maximize $f(a,b)=\frac{1}{2}a^2+4b^2-4a+8c^2$ subject to the constraint : $g(a,b)=\frac{1}{2}a+3b+c-25$ where $g(a,b)=0$
I write the lagrangian
$$L=\frac{1}{2}a^2+4b^2-4a+8c^2-\delta(\frac{1}{2}a+3b+c-25)$$
And solve. I get the optimum point at : $(12,6,1,16)$
In order to check whether that's a max or a min I compute the Hessian matrix of $f$ Which is
$$H_f= \begin{bmatrix} 1& 0& 0 \\ 0 & 8 & 0 \\ 0 & 0 & 16\end{bmatrix}$$
I see that $H_f$ is positive definite. This would imply that the critical point $(12,6,1,16)$ is a minimum. right?
Or does that imply that the critical point of the f is a minimum, not the critical point of the lagrangian. So Do I have to check the Hessian matrix of the lagrangian instead?
If that's not the case, then I constructed the lagrangian in a way that I would be maximising $f$ since $-\delta$ is the parameter. Then what is going on?