Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 11 / a / i / Solution

For a twice continuously differentiable function, a finite second-order Taylor expansion about an interior point iswhere $A=f_{xx}$, $B=f_{xy}$, $C=f_{yy}$ at that point. A differentiable interior extremum requires $f_x=f_y=0$. For a strict minimum guaranteed by the quadratic term, the Hessian matrix must be positive definite. Completing the square gives $A{a}^2+2Bab+C{b}^2=A(a+Bb/A{)}^2+(AC-B^2)b^2/A$. Thus sufficient nondegenerate minimum conditions areIf $D=0$, the second-order test is inconclusive; for example $x^4+y^4$ has a strict minimum at zero despite a zero Hessian. The finite Taylor argument only needs $C^2$ regularity. The source's infinite Taylor representation additionally needs suitable analyticity: a smooth flat function, such as $e^{-1/(x^2+y^2)}$ extended by zero at the origin, has every Taylor coefficient zero there but is positive nearby.