Section 2-4:  Finding Maximums and Minimums
 
 
When a quadratic function is used in finding a maximum or minimum value, you can use the fact that the vertex happens at x = -b/2a!!
 
 
Sample Problem
A rectangular  pen is constructed using one side of the house as a side of the pen.  You have 100 feet of fencing for the other three sides.  Find the dimensions of the greatest size area inside the pen.
Solution!!
 
Make a diagram!!
 
A(x) = x(100 - 2x) = 100x - 2x2 = -2x2 + 100x
The maximum happens at x = -b/2a = -100/(-4) = 25
The dimensions are 25 ft by 25 ft
The greatest area is 25(100 - 50) = 25(50) = 1250 sq ft!!

For a cubic function, we will use the functions on the Ti-82 to graph the function and then find the maximum point using the calc function on the calculator to approximate the answer.
 
 
Sample Problem
 
Squares are cut from the corner of  a rectangular piece of cardboard with dimensions 8 in by 12 in.  The sides will be turned up to form a box with no top.  Find the maximum volume of the box.
 
Solution:  Draw a picture!!
 
V(x) = x(12 - 2x)(8 - 2x)
Type this function into the TI-82 calculator!
Using the calc button, estimate the maximum point on the graph!!  You should get ( 1.6, 67.6) rounded to tenths.
This means that the corner should be cut to 1.6 inches and the maximum volume will be about 67.6 cubic inches!!

Sample #3
A manufacturer has 100 tons of a product that he can sell now for a profit of 5 per ton.  For each week he delays shipment, he can produce an additional 10 tons.  Unfortunately, for each week he delays, the profit decreases 25 cents a ton.  When should he ship to maximize his profit and what is the maximum profit?
Solution:  Let x = number of weeks to delay
 
   Number of tons   Profit in dollars
          Now            100    100(5) = $500
      In x weeks       100 + 10x         5 - .25x
P(x) = (100 + 10x)(5 - .25x)
                = 500 + 25x - 2.5x2 using foil
The maximum value happens at x = -b/2a = -25/-5 = 5
He should sell in 5 weeks!!
The profit will be 500 + 25(5) - 2.5(5)2 = $562.50

That's all for this section!!  On to the next!