Question

Find the absolute minimum and absolute maximum values of f on the given interval. f(x) = x − ln 8x, 1 2 , 2

Answer 1

First step: Find derivative
Second step: Find critical numbers that occur between the endpoints.
Third Step: Plug in critical numbers that are between the endpoints and also plug in endpoints into f.

Answer 2

Can find the derivative?

Answer 3

the endpoints are given

Answer 4

I will walk you through it.
But I would like you to first find f'(x)

Answer 5

(x)'=?
(ln(8x))'=?

Answer 6

And I think you mean the interval is [1/2,2]

Answer 7

1/2 is the left endpoint
2 is the right endpoint

Answer 8

Do you know (x)'=?

Answer 9

What is the slope of the line y=x?

Answer 10

\[\frac{d}{dx}(\ln(8x))=\frac{d}{dx}\ln(8)+\frac{d}{dx}\ln(x)=?\]

Answer 11

well it looks like you know the derivative of x
but not the derivative of constant or ln(x)

Answer 12

\[\frac{d}{dx}c=0 \text{ where c is a constant } \\ \frac{d}{dx}\ln(x)=\frac{1}{x}\]

Answer 13

\[y'=1-\frac{1}{x}\]
Find the critical numbers now.

Answer 14

You need to find when y'=0 and
when y' dne (we only care about the values that make y' dne that actually exist in the domain of the original function)

Answer 15

So I find it is always easiest to combine fractions
y'=(x-1)/x

Answer 16

no you find when y'=0 and when y' dne

Answer 17

We have a fraction for y'
All you have to do to find when y'=0 is to find when the top is equal to 0
To find when y' dne you find when the bottom is 0 (again here we only care about numbers that actually happen in our domain)

Answer 18

no

Answer 19

We are simply finding the critical numbers right now

Answer 20

critical numbers could be min or max or maybe not even either

Answer 21

Right. We also need to consider the critical numbers that we have yet to find.

Answer 22

Let me know when you have the equations and I will check to see if you have the right critical numbers.

Answer 23

y'=0 when top=0
y' dne when bottom=0
and you have y'=(x-1)/x

Answer 24

Just summarizing what we have already said

Answer 25

@garyforest ...
have you set the top equal to 0 yet?

Answer 26

I don't mean to be impatient but you should have already been able to solve x-1=0

Answer 27

Right.

Answer 28

so x=1 is a critical number
and it is a critical number that occurs in between the endpoints

y' dne at x=0 but we don't care about x=0 anyways because ln(0) doesn't even exist
so x=0 doesn't even occur in our domain
also it doesn't even occur in the given interval

Answer 29

so check x=1/2 , check x=1 , check x=2

Answer 30

\[f(x)=x-\ln(8x)\]

Answer 31

plug each number into your initial

Answer 32

and the smallest number will be your absolute min
the largest number will be your absolute max

Answer 33

I will leave this part to you
because I must go now