Maximum or Minimum Values of Polynomial Functions
Problem 1:
Two numbers have a difference of 6. Find their minimum possible
product.
-
Set up variable expressions to represent the various quantities used in
the problem.
Let x = one number
and y = the other number
-
Set up a variable expression representing the quantity to be minimized.
P(x,y) = xy
-
If possible, describe one of the variables in terms of the other.
Since y - x = 6, then y = 6 + x
-
Substitute this expression into the product function.
P(x) = x (6 + x) = 6x + x2
-
Graph this function in your calculator and use 2nd Calc Min to find
the minimum point.
The point is (-3, -9).
-
So the minimum possible product is -9. (The y-value names
the product)
Problem 2:
An open box is to be formed by cutting squares from a square sheet of metal
10
cm on a side and then folding up the sides. Find the size of the
square that maximizes the volume, and give the maximum volume.
-
Draw a picture of the problem, if possible.
-
Set up variable expressions to represent the various quantities used in
the problem.
Let x = the size of the square to be cut out of
the corners which is the height of the box
formed by folding up the sides.
10 - 2x = the length of the box formed by folding up
the sides.
10 - 2x = the width of the box formed by folding up the
sides.
-
Set up a variable expression representing the quantity to be minimized.
V(x) = x(10 - 2x)2
-
Think about the domain and range of this function so the viewing window
in the calculator can be set appropriately.
0 < x < 5 V(x)
> 0
-
Graph this function in your calculator and use 2nd Calc Max to find
the maximum point.
The point is (1.667, 74.074).
So the size of the square to be cut out is 1.667 cm, and
the maximum volume possible is 74.074 cu. cm.
Problems
(there are no questions here yet)
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