Finding saddle point from the critical points

Question

If $(4,0)$ and $(0,-1/2)$ are critical points of the function:
$$f(x,y)=5-(alpha +beta)x^2 +beta y^2+(alpha+1)y^3+x^ 3,$$where $   alpha,beta in mathbb R$ then prove that $(4,-1/2)$ is a saddle point of $f$.
I calculated the Hessian but I am unable to prove that the Hessian is indefinite at the given point $(4,-1/2)$. The problem is I am not able to utilize the given critical points to any advantage to arrive at the desired conclusion. Please help.

enzotib · Accepted Answer

The first order derivatives, evaluated in the two know critical points are
begin{alignat}{2}
&frac{partial f}{partial x}(4,0)    &&= 48-8(alpha+beta),\
&frac{partial f}{partial y}(4,0)    &&= 0,\
&frac{partial f}{partial x}(0,-1/2) &&= 0,\
&frac{partial f}{partial y}(0,-1/2) &&= frac{3}{4}(alpha+1)-beta,\
end{alignat}
and to be zero, we should solve
begin{align}
&48-8(alpha+beta)=0,\
&frac{3}{4}(alpha+1)-beta=0
end{align}
from which we obtain $alpha=beta=3$.
With this values of the two parameters, it is easy to show that the Hessian determinant evaluated in the point $(4,-1/2)$ is $-72.$

Finding saddle point from the critical points

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