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Finding saddle point from the critical points

Mathematics Asked by Lawrence Mano on January 22, 2021

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.

One 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 $-72.$

Correct answer by enzotib on January 22, 2021

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