Relative maxima/minima: a quick review
A company manufactures two products A and B that sell for $10 and $9
per unit respectively. The cost of producing x units of A and y units of B
is
400 + 2x + 3y + 0.01(3x 2 + xy + 3y 2 ).
Find the values of x and y that maximize the company’s profits.
A. (100, 80)
B. (120, 90)
C. (120, 80)
D. (80, 120)
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
1/4
Absolute maxima/minima on closed, bounded sets
Let R be a closed, bounded set in the plane.
The word “bounded” means that R is contained within a fixed ball
centered at the origin.
The word “closed” means that the boundary of R is contained in R.
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
2/4
Absolute maxima/minima on closed, bounded sets
Let R be a closed, bounded set in the plane.
The word “bounded” means that R is contained within a fixed ball
centered at the origin.
The word “closed” means that the boundary of R is contained in R.
Statement of the problem
Let f be continuous on a closed, bounded set R in the plane. Find the
largest value (or maximum) and the smallest value (or minimum) that f
attains on R. In other words, find points (a, b) and (c, d) in R such that
f (c, d) ≤ f (x, y ) ≤ f (a, b)
Math 105 (Section 204)
for all (x, y ) in R.
Multivariable Calculus – Extremization
2011W T2
2/4
Procedure for finding absolute maxima/minima
Step 1
Find the values of f at all critical points in R.
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
3/4
Procedure for finding absolute maxima/minima
Step 1
Find the values of f at all critical points in R.
Step 2
Find the maximum and minimum values of f on the boundary of R.
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
3/4
Procedure for finding absolute maxima/minima
Step 1
Find the values of f at all critical points in R.
Step 2
Find the maximum and minimum values of f on the boundary of R.
Step 3
The largest (respectively smallest) function value found in steps 1 and 2 is
the absolute maximum value (respectively absolute minimum value) of f
on R.
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
3/4
An example
Find the absolute maximum and minimum values of the function
p
f (x, y ) = x 2 + y 2 − 2x + 2
on the closed half-disk
R = {(x, y ) : x 2 + y 2 ≤ 4 with y ≥ 0}.
A. max = 6, min = 1
√
B. max = 10, min = 2
√
C. max = 10, min = 1
D. max = 6, min = 2
Math 105 (Section 204)
Multivariable Calculus – Extremization
2011W T2
4/4