Problem :
Find 2 positive numbers whose product is 25 and whose sum is a minimum.
| S | =  + x; domain = (0,∞) |
||
| S'(x) | =  + 1 |
||
| S'(x) = 0 when x = 5 |
S''(x) = 
|
Problem :
What is the maximum value of
f (x) = x
4 -8x
2 - 3
on the interval
[- 3, 3]
?
Problem :
A shepherd wishes to build a rectangular fenced area against the side of a barn. He has 360 feet
of fencing material, and only needs to use it on three sides of the enclosure, since the wall
of the barn will provide the last side. What dimensions should the shepherd choose to
maximize the area of the enclosure?
Problem :
Find the point on the graph of
y = x
2
that is the smallest distance from the point
(0, 6)
.
We want to minimize this function.
In practice, minimizing the distance is the same as minimizing the square of the distance,
and because working with square roots can become complicated, we will choose here to
minimize the function
D
2
, which is the square of the distance.
So,
D
2 = (y - 6)2 + x
2
.
Constraint:
y = x
2
.
Substituted objective:
D
2 = (x
2 -6)2 + x
2
.
| (D 2)'(x) = 2(x 2 -6)2(2x) + 2x | |||
| = 4x 3 - 24x + 2x | |||
| = x(4x 2 - 22) | |||
(D
2)'(x) = 0 atx = 0 and atx = ±
|
|||
| Now use the second derivative: | |||
| (D 2)''(x) = 12x 2 - 22 | |||
| (D 2)''(0) < 0, so it is a local max. | |||
| (D
2)''() > 0, so it is a local min. |
|||
| (D
2)''(- ) > 0, so it is a local min also. |
,
)
and
(- ,)
on the graph of
y = x
2
are both
the exact same distance from the point
(0, 6)
. To see that these local minima
are also at the absolute minimum distance, consider the following diagram.
.
