Let $f : R^2 → R$ a function of class $C^2$ near to $(0,0)$. If it is known by the second derivative test that f(x,y) has a local maximum in $(0,0)$, show that exists some $ k \in R$ such that $g(x,y) = f(x,y) + k(x^2 + y^2)$ has a local minimum in $(0,0)$
My approach:
$H f(0,0) = $ $\begin{bmatrix} \frac{\partial^2 f}{\partial x^2}(0,0)& \frac{\partial^2 f}{\partial y\partial x}(0,0)\\ \frac{\partial^2 f}{\partial x\partial y}(0,0)& \frac{\partial^2 f}{\partial y^2}(0,0) \end{bmatrix}$ and due to $f$ is of class $C^2$ then $\frac{\partial^2 f}{\partial x\partial y}=\frac{\partial^2 f}{\partial y\partial x}$
then we can write $Hf(0,0)= \begin{bmatrix} a& b \\ b& c \end{bmatrix}$
If $f$ has a local maximum then the quadratic form definited by to $Hf$ is negative definite i.e., $ax^2 +2bxy+cy^2 <0$
On the other hand, $Hg(0,0)= \begin{bmatrix} a+2k& b \\ b& c+2k \end{bmatrix}$ and in order to prove that $g$ has a minimum we must show that exists some $k$ such that $(a+2k)x^2 +2bxy+(c+2k)y^2 >0$
I'm stuck here, don't know how can I compute $k$. Is it right or is there another way to solve it?