Finding the minimum value of $1/2 (x_{1}^2+x_{2}^2)-x_{2}b_{2}$

Question

I am trying to find the minimum value of the following:

$1/2 (x_{1}^2+x_{2}^2)-x_{2}b_{2}$

I know this is equal to:

$1/2 ([x_1 x_2] Id [x_1 x_2]^T )-[x_1 x_2] [0 b_2]^T$

To find minimum we have to consider the positive definite matrix of $x^T Ax$ When I am calculating the Hessian I get a 2x2 matrix of 0's.

I think I have made a mistake so can anyone help troubleshoot here?

Answer 1

You do have this as $$\frac12 \left( x_1^2 + (x_2-b_2)^2-b_2^2 \right)$$

from where the minimum is obvious.