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18.02 Multivariable Calculus
Fall 2007
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View Full Document SD. Second Derivative Test
1.
The Second Derivative Test
We begin by recalling the situation for twice differentiable functions f (x) of one variable.
To find their local (or "relative") maxima and minima, we
1. find the critical points, i.e., the solutions of fl(x)
=
0;
2. apply the second derivative test to each critical point xo:
f1I(xo)
>
0
+
xo is a local minimum point;
fI1(xo)
<
0
+
xo is a local maximum point.
The idea behind it is: at xo the slope fl(xo)
=
0; if fl1(xo)
>
0, then fl(x) is strictly
increasing for x near xo, so that the slope is negative to the left of xo and positive to the
right, which shows that xo is a minimum point. The reasoning for the maximum point is
similar.
If fl1(xo)
=
0, the test fails and one has to investigate further, by taking more derivatives,
or getting more information about the graph. Besides being a maximum or minimum, such
a point could also be a horizontal point of inflection.
The analogous test for maxima and minima of functions of two variables f (x, y) is a
little more complicated, since there are several equations to satisfy, several derivatives to be
taken into account, and another important geometric possibility for a critical point, namely
a
saddle point.
This is a local minimax point; around such a point the graph of f (x, y)
looks like the central part of a saddle, or the region around the highest point of a mountain
pass. In the neighborhood of a saddle point, the graph of the function lies both above and
below its horizontal tangent plane at the point. (Your textbook has illustrations.)
The secondderivative test for maxima, minima, and saddle points has two steps.
fx(x, Y)
=
0,
1. Find the critical points by solving the simultaneous equations
fy(x,y)
=
0.
Since a critical point (xo, yo) is a solution to both equations, both partial derivatives are
zero there, so that the tangent plane to the graph of f (x, y) is horizontal.
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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.
 Spring '08
 Auroux
 Derivative, Multivariable Calculus

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