2.4 Finding Maximums and Minimums of Polynomial Functions

Objective: I will be able to determine maximums and minimums of polynomial functions to solve real world problems.

Quadratic Functions

Where do you think the maximum and minimum values of a quadratic are?

The vertex is the maximum (or minimum) of a quadratic. When would it be the minimum? The maximum?

Who remembers the equation to find the vertex of a quadratic?

Why do we care about maximum/minimum of quadratic functions?There are many real world situations where this is modeled.

Area (Example #1)

x must be less than 30

x= 15 so the maximum dimensions are 15 x 30 (450m2)

A rectangular dog pen is constructed using a barn wall as one side and 60m of fencing for the other three sides. Find the dimensions of the pen that give the greatest area.

barn wall

xx

60 – 2x

Income/Revenue (Example #2)

The publisher of a magazine that has a circulation of 80,000 and sells for $1.60 a copy decides to raise the prices of the magazine because of increased production and distribution costs. By surveying the readers of the magazine, the publisher finds that the magazine will lose 10,000 readers for each $0.40 increase in price. What price per copy maximizes the income?

Let x = the # of +$0.40 price intervals to obtain maximum income

Income = # of copies sold x price per copy

Cubic functions

What do you think the maximum and minimum values of a cubic function are?

All cubic functions (except those with double or triple roots) have either a “peak” or “valley”

We call this the “local” maximum/minimum because it technically isn’t the highest/lowest y-value in the graph (remember the end behavior of cubic functions!)

Volume (Example #3)

Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions of 6 in. and 10 in. The metal is then folded to make an open-top box. What is the maximum volume of such a box?

The dimensions of the box are:

height = x; length = 10 – 2x; width = 6 – 2xx must be between 0 and 3

x ~ 1.2 so the maximum volume is ~32.8in.3

You try!• Shoe Carnival has found that they can save millions of

dollars a year by minimizing the amount of cardboard they use to construct their shoe boxes. The company is considering using a sheet of cardboard with dimensions of 6in and 10in with a square cut out. However before Shoe Carnival purchases these in bulk, they want to know the maximum volume of such a box in order to figure out if the majority of their shoes will fit in it. What is the maximum volume and is this cost saving measure really going to be effective?

Around the World (no, not the Daft Punk song)•Get into your groups•Min/Max problems are posted around the room- please try to complete all of them• Each group starts at a different one•Move to another problem once you finish- no more than 2 groups at a problem at a time!