2.6 Extreme Values of Functions-max-min

-standard form-applications of extremes

Extreme value

• The largest or smallest value of the function on some interval.

• Extreme values are important in several applications. I.e. profits, production, costs, speed.

There are two types of extremes

• Maximum – highest point on an interval• Minimum – lowest point on an interval

Interval : [-4.5, 5.5] Interval : [-5.2, 4.9]

Standard form of a quadratic function

2( )f x ax bx c Quadratic function

Written in STANDARD FORM

2( ) ( )f x a x h k

2( ) 5 30 49f x x x 25( 6 ) 49x x 25( 6 9) 49 5 9x x

25( 3) 4x

2( ) 5( 3) 4f x x

When the quadratic is in this form2( ) ( )f x a x h k

There will be a minimum if a>0 (graph will be concave up)

There will be a maximum if a<0 (graph will be concave down)

The extreme point will be at (-h, k)

• This is a lot of work if the function is given in the form because we have to complete the square.

So for our uses in this class, we will only use this rule of extreme values if the function is presented to us in standard form already.

2( )f x ax bx c

2( ) ( )f x a x h k

So what about the quadratics not in standard form???

2( )f x ax bx c If a < 0, maximumIf a > 0, minimum

We can find the x value by using the following formula

2bxa

Substitute x back into ax2+bx+c to find the y

value of the max or min.

2( ) 3( 4) 6f x x

2 1( ) ( 2)2

f x x

2( ) 2 4 5f x x x

2( ) 4 12f x x x

Most cars get their best gas mileage when traveling at a modest speed. The gas mileage M for a certain new car is modeled by the function.

where s is the speed in mi/h and M is measured in mi/gal. What is the car’s best gas mileage, and at what speed is it attained?

21( ) 3 3128

M s s s

Speed

mi/g

• Maximums and minimums are often best used for OPTIMIZATION where you are looking to make something as big as possible or make something as small as possible.

• i.e, profits, resources, areas, distances, (volumes in calculus)

• Two numbers add up to 20. What are the two numbers that maximize the product?

• Let 1st # = x• Let 2nd # = 20-x• What is the product of these two numbers?

• A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not want a fence along the river. What are the dimensions of the field that would create the largest area?

• A farmer has 1800 m of fencing with which to build three adjacent rectangular corrals, as shown. Find the dimensions so that the total enclosed area is as large as possible.

• A rancher with 750 ft of fencing wants to enclose a rectangular area then divide it into four pens with fencing parallel to one side of the rectangle. Find the largest possible area for the 4 pens, find the area of one pen if all 4 are equal in size.

• The perimeter of a rectangle is 2000 cm. Express the area of the rectangle in terms of its width w.

• Among all rectangles having a perimeter of 10 ft, find the dimensions (length and width) that maximizes the area.–Need the formula for perimeter–Need the formula for area

–We will solve one equation for a variable, and then substitute back into the other equation.

– How do you know which equation to solve for? Well what information are you given??? That should help.

2 2P l w A lw

• Suppose that you have 600 m of fencing with which to build 2 adjacent rectangular corrals. The 2 corrals are to share a common fence on one side. Find the dimensions x and y so that the total enclosed area is as large as possible.

yx x

y

• You run a canoe-rental business on a small river in Ohio. You currently charge $12 per canoe and average 36 rentals a day. An industry journal says that, for every fifty-cent increase in rental price, the average business can expect to lose two rentals a day. Use this information to attempt to maximize your income. What should you charge?

Price Hikes Price per rental Number of rentals

Total Income/Revenue

None $12.00 36 12.00(36)

1 $12.00 + 1(0.50) 36 – 1(2) 12.50(34)

2 $12.00 + 2(0.50) 36 – 2(2) 13.00(32)

3 $12.00 + 3(0.50) 36 – 3(2) 13.50(30)

x $12.00 + x(0.50) 36 – x(2)

• Squares with sides of length x are cut from the corners of a rectangular piece of sheet metal with dimensions of 8 inches and 11 inches. The metal is then folded to make an open-top box. What is the maximum volume of such a box?

xx

11 in

8 in

*key elements

V=lwhDo you have a method for finding the length, width, and height?

• Now with the previous slide we can look at the expressions for length, width, and height and justify to ourselves that x has to fall somewhere in the interval of (0,4) because we want our dimensions to be positive numbers. **distance has to be positive**

• Because this polynomial becomes a cubic, we cannot find the maximum or minimum using

x=(-b/2a) so we will revert to the use of technology to help find the max or min.-Calculus would allow us to find the maximum and minimum without the use of technology.

Homework

• Pg. 71– #s 1-3, 9, 10, – Pg. 93 7