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 In singlevariable calculus, you used the second derivative to determine whether a critical point of a function was a maximum or a minimum. For multivariable calculus, we also have a secondderivative test, though it is more complicated.
SecondDerivative Test: Suppose $f\colon\R^2\to\R$ has continuous second partial derivatives at a critical point $(x_0,y_0)$ of $f$. Let $$ D = f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)  \bigl(f_{xy}(x_0,y_0)\bigr)^2 $$ Then:
 If $D > 0$ and $f_{xx}(x_0,y_0) > 0$ then $f$ has a relative minimum at $(x_0,y_0)$.
 If $D > 0$ and $f_{xx}(x_0,y_0) < 0$ then $f$ has a relative maximum at $(x_0,y_0)$.
 If $D < 0$ then $f$ has a saddle point at $(x_0,y_0)$.
 If $D = 0$ then the test tells us nothing (no conclusion can be drawn).
Consider the function $f(x,y) = xy  x^3  y^2$.
 Locate all the critical points of $f$.
 Use the second derivative test to determine the type of critical point in each case.
 Show that the fourth condition above is necessary by producing examples of functions where $(0,0)$ is a critical point and $D=0$ at $(0,0)$ but where:
(Hint: Try to do something like our singlevariable examples from class.)
 The function has a minimum at $(0,0)$.
 The function has a maximum $(0,0)$.
 The function has a saddle $(0,0)$.
 The function has none of the above $(0,0)$.

