I have the following equation:
$$ f(x,y) = x^2 + cy^2 + 2xy $$
I want to find the values of c for the function $f(x,y)$ has a minimum. However I have to solve it by writing the function in the following form:
$$ f(x,y) = \begin{bmatrix} x \\ y \end{bmatrix} ^T A \begin{bmatrix} x \\ y \end{bmatrix}$$
I generally solve the equations by differentiating by x and y and equating to zero, then forming a Hessian matrix and then checking if my point(s) are a minimum or not. However, I am very unfamiliar to solving it in the form above. Could someone explain the relevance of the form to finding the minimum.
Best Answer
We have: $f(x,y) = (x+y)^2 + (c-1)y^2$. Thus if $c = 1$, clearly the minimum is $0$ when $x = -y$. If $c \neq 1$, then either $c < 1$ or $c > 1$. For $c < 1$, take $x = -y$ and thus $f(-y,y) = (c-1)y^2$ has no minimum. For $c > 1$, again $f(-y,y) = (c-1)y^2 \ge 0$ and has a minimum of $0$ when $y = 0$ and in this case $x = -y = 0$ as well. In brief, the minimum occurs for $f$ when $c \ge 1$.