
12.7. Extreme Values
R2
Local extrema occur at critical points where f (x)  0 or
f (x) is undefined.
Tests
1. First derivative test
2. Second derivative test

R3
r
Local extrema occur at critical points where f (x,r y)  0 or
f (x, y) is undefined. Note: when f (x, y)  0 , fx = 0
and fy = 0 simultaneously.

A saddle point is a point (a,
r b, f(a, b)) whose tangent plane is
horizontal ( f (x, y)  0 ) and f has both higher and lower
function values on any region containing (a, b).

Local extrema = local maxima, local minima, and saddle points.
Second Partials Test
r
Let z = f(x, y) and suppose f (a,b)  0 . Define D(a, b), “the
discriminant of f”, as follows:
D(a, b) = fxx(a, b)fyy(a, b) – (fxy(a, b))2

1. If D(a, b) > 0 and fxx(a, b) > 0, then f has a local minimum
at (a, b)
2. If D(a, b) > 0 and fxx(a, b) < 0, then f has a local maximum
at (a, b)
3. If D(a, b) < 0, then f has a saddle point at (a, b)
4. If D(a, b) = 0, then it’s inconclusive
Ex. f(x, y) = x4 + y3 + 32x – 9y
Ex. f(x, y) = xy2 – 6x2 – 3y2
Do: Find all local extrema for f(x, y) = 6x2 - 2x3 + 3y2 + 6xy
Absolute Maxima and Minima
If f is a continuous function on a closed and bounded region R,
then f has an absolute maximum and absolute minimum. The
absolute maximum and minimum will always be at a critical
point or on the boundary.
R2
R3
Steps:
1.
2.
3.
4.
Find the critical points and list them.
List the end points or corners
Find the critical points on the boundaries
Find the function values: the largest is the absolute
maximum and the smallest is the absolute minimum.
Ex. Find the absolute maximum and minimum for
f(x, y) = x4 + y3 + 32x – 9y on the region -2 ≤ x ≤ 0 and -2 ≤ y ≤ 0
Ex. f(x, y) = xy2 – 6x2 – 3y2 on the triangular region with
vertices (1, 1), (-1, 1), and (-1, -3).
Ex. f(x, y) = xy2 – 6x2 – 3y2 on the region bounded by y = x2 and
y = 1.
Ex. f(x, y) = x + y2 on the region bounded by x2 + y2 ≤ 1.