Mathl 15

Notes

Finding Maximum/Illinimum

Values

Example:

Finding &e

sales

level

for

maximum

profit given

the equation

for

a

profit

function.

If

the sales

level

is

fizctional,

round

up & down to

check the

value

of

the

profit

at both

values.

Normally, in business,

we

do

not sell

a fraction

of

an item.

Given:

P(x):

-x3+77x2-150x+200 where 0<x<70.

Method

1: Using

your

calculator.

First, code

in the

profit

fimction

using the

y:

menu.

Make

6\

Sure

that

the first

negative sign

is from the bottom

of

your

calculator.

(y

Make sue

your

table

is set up

properly.

Go to

the

2od window menu For the independent variable

it shoutd

be set

at

ASK &

for the

dependent

variable it

should

be set at Auto. Keep this setting though out the

entire

course.

Before

setting

a

viewing

window,

get

some idea

of the values involved by

going

to

2"d

graph.

A

table

for the x

&

y

values

should

be displayed.

The

y

values

are

the P(x) values for each

x

input.

Since

x

is resEicted

betrveen 0 &ll,input

values starting at 0 in intervals up to 70 and

take a

note of

the

magnitude

of the

numbers involved. Do

not skip too many since

you

could miss the

maximum

value.

You

will notice that

the

values

for P(x)

will range from 200 to over

60000 then

start

to dectease.

So,

for

the

x window,

use a

low

value

(min

)C) of 0 and

a

high value

(ma:r

x)

of

70.

For

the

y

window,

use

a

min

of 0

and a

nrax of 80000

(to

be

safe).

Press

graph

to check

your graph.

The manimum

(hrghe$ poin$

should

be

visible. If

no!

adjust

your

window

(y

range)

nntil

it is clearly

visible.

Press

2d

Calc, then

go

to menu 4, enter. The

graph

should be visibte. Press trace.

A marker

on

the

ctrve

should

be

visible

& can be

moved along the curve using the

right

&

left amows.

The calculator will

prompt

you

for

a left

guess

for

x. Enter any

value for x

on the

left

side

of the highest

point

enter.

Then

do the same

for

any value

for x

on the

right side,

enter. The calculator

will

then

prompt you

for

a

guess.

Enter any x

value

between

your

left

&

right

values. Then the

answer for x at the maximum will be displayed

at the bottom along with

its

y

value.

If all

prccedures

are followed

correctly, it

should

be x:50.34. Check x=5I using

the tablq

that

gives

aprofitof

$60,176.

At te50, the

profit

is

$60,200.

So, x:50

produces

the maximum

profit

of

$60,200.

The

value

for the

profit

at the

right end

point"

l,:70,

is

$24,000.

So, that

value for

x can be eliminated.

So can:<=0.

Method

2: The derivative

method. Find the values for x for which P'(x):Q.

Since this is

a

quadratic

equation

(does

not factor), the

quadratic

formule from algebra must be used. Take

the

positive

value

for

x.

Solve:

-3x2

+154x-150{

or

3x2

-154x+150E0,

by multiplying both sides

by a negative l.

x

=

{-rs4)+Jt-ts7'H3l[5di

all divided

by 2Q).

Note:

If the

profit

firnction

is 46 degree or higher,

the calculator

method must

be used.

So, know

it

well.

Note:

Below is a sketch of thc

situation aod should help

you

understand

the

problem

better. [n

most

cases,

a sketch

is not required

(onty

if asked).