f(x)> f (x 0 )) holds. The determination of relative maximum and minimum values of
a differential function y=f(x)over an interval (a, b)consists of
1. Determining the critical points where f′(x) = 0 and then testing these critical
points.
2. Testing the boundary points x=aand x=b.
The second derivative test for relative maximum and minimum values states
that if x 0 is a critical point, then
1. f(x)has the maximum value f(x 0 )if f′′(x 0 )< 0 (i.e., curve is concave downward
if the second derivative is negative).
2. f(x)has a minimum value f(x 0 )if f′′(x 0 )> 0 (i.e., the curve is concave upward if
the second derivative is positive).
The above concepts for the relative maximum and minimum values of a func-
tion of one variable can be extended to higher dimensions when one must deal with
functions of more than one variable. The extension of these concepts can be accom-
plished by utilizing the gradient and directional derivatives.
In the following discussion, it is assumed that the given surface is in an explicit
form. If the surface is given in the implicit form F(x, y, z) = 0 ,then it is assumed that
one can solve for zin terms of xand yto obtain z=z(x, y ).By a delta neighborhood
of a point (x 0 , y 0 )in two-dimensions is meant the set of all points inside the circular
disk
N 0 (δ) = {x, y |(x−x 0 )^2 + (y−y 0 )^2 < δ^2 }.
The function z(x, y ),which is continuous and whose derivatives exist, has a relative
maximum at a point (x 0 , y 0 )if z(x, y )< z(x 0 , y 0 )for all x, y in a some δneighborhood
of (x 0 , y 0 ).Similarly, the function z(x, y )has a relative minimum at a point (x 0 , y 0 )if
z(x, y )> z(x 0 , y 0 )for all x, y in some δneighborhood of the point (x 0 , y 0 ).Points where
the surface z=z(x, y )has a relative maximum or minimum are called critical points
and at these points one must have
∂z
∂x = 0 and
∂z
∂y = 0
simultaneously. Critical points are those points where the tangent plane to the
surface z=z(x, y )is parallel to the x, y plane. If the points (x, y )are restricted to a
region Rof the plane z= 0,then the boundary points of Rmust be tested separately
for the determination of any local maximum or minimum values on the surface.
