There is a question im trying to solve, but im not sure im doing the right thing.
"Let $p(x) = \sum_{j=0}^na_jx^j$ a polynomial without multiple roots, then all critical points of $f(x,y)=cosh(x)+y\cdot p(x)$ are saddle points."
My task is to proof or disproof this statement.
I tried this:
Write $p(x)=(x-r_1)(x-r_2)\ldots(x-r_n)$, where each $r_i$, $i=1\ldots n$, are the roots of $p(x)$. Considering $v=(v_1,v_2)\in\mathbb{R^2}-\{(0,0)\}$ and the Hessian form $H(x,y)v^2 = \frac{\partial^2f}{\partial x\partial x}v_1v_1+2\frac{\partial^2f}{\partial x\partial y}v_1v_2+\frac{\partial^2f}{\partial y\partial y}v_2v_2$
if $H$ is indefinite for all critical points, then the statement is true, otherwise is false.
So, computing the partial derivates and the critical points i found $H(r_i,y_i)v^2=\big(cosh(r_i)+y_i(2a_2+\ldots+n(n-1)a_nr_i^{n-2})\big)v_1^2+(a_1+2a_2+\ldots+na_nr_i^{n-1})v_1v_2$
where the points $(r_i,y_i)$, $i=1\ldots n$, are the critical points($r_i$ is a root of $p(x)$).
I think i can prove that the statement is true by choosing $v_1,v_2$ properly and analyzing the signals, but im not sure how and im not even sure if this is right. Any help is welcome, thanks!