1. 4.7
Optimization Problems

2. Steps to follow for max and min problems.
(a) Draw a diagram, if possible.
(b) Assign symbols to unknown quantities.
(c) Assign a symbol,Q, to the quantity to
be maximized or minimized.
(d) Express Q in terms of the other symbols.
(e) Eliminate all but one unknown symbol,
say x, from Q.

3. (f) Find the absolute maximum or minimum
of Q = f (x).
(iii) Use the first derivative
(iv) Does the absolute max or min

4. 1. Example:
Solution:
=˃p(x)=x(x-100)

5. (i) Domain of P(x):
all real numbers.
(ii) Critical numbers of P(x)
P′(x)=
(iii) First Derivative Test
P′(x)

6. P(50) is an absolute min value.

7. Graph:

8. 2. Example:
A farmer wants to fence a
rectangular enclosure for his horses
and then divide it into thirds with fences
parallel to one side of the rectangle. If
he has 2000m of fencing, find the area
of the largest rectangle that can be
enclosed.
Solution:

9. (i) Domain of A(l):

10. (ii) Critical numbers of A(l)
By the first derivative test we can show that A(500) is the max value of A,

11. When l=500,
max area of the rectangle =

12. Graph:

13. x=y2 that is closest to the poit (0,3).
3. Example:
x=y2 that is closest to the poit (0,3).
Let D denote the distance from (0,3) to
any point (x,y) on the parabola.

14. D(y)=
Find the value of y which makes D(y)
(i) Domain of D(y):
all real numbers > 0.

15. (ii) Critical numbers:
D′(y)

16. (iii) First Derivative Test
D′(y)
By the first derivative test D(y)

17. Graph:

18. 4. Example: A right circular cylinder is
Solution:

19. (i) Domain of V(y):

20. (ii) Critical numbers of V(y)
2π r2 -3y2
By the first derivative test

22. 5. Example: Mountain Beer sells its beer
in an aluminum can in the shape of a right-
circular cylinder. The volume of each can
What should the dimension of
the can be in order to minimize the amount
of aluminum used?
Solution:
and

23. r h
Surface Area,

24. S(r)=
(i) Domain of S(r)
(ii) Critical numbers of S(r)
S′(r)=

25. (iii) First Derivative Test
S′(r)=0 =˃
(iii) First Derivative Test
S′(r)=
S′(r)
S(r)
S(r) has its absolute minimum value

27. Graph: