$($Local Max$/$Min$)*$

Consider a function $f(x,y)=-(x^2-1{)}^2-(x^{2}y-x-1{)}^2.$ Which of the following is true?

$(1)f$ has only one stationary point.

$(2)-1$ is a local maximum of $f.$

$(3)f$ has a local maximum at euch critical point.

$(4)f$ has a local minimum at each critical point.

$(5)$ At each stationary point, the hessian matrix of $f$ is nether negative definite, nor positive definite.

$(1)$

$(2)$

$(3)$

$(4)$

$(5)$