Given that f(1) = 3 and 2 less than or equal to f'(x) less than or equal to 3 for all x, what is the smallest possible value for f(4)?

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- July 27th 2006, 12:21 PMNimmyCalculus I Homework Help part II
Given that f(1) = 3 and 2 less than or equal to f'(x) less than or equal to 3 for all x, what is the smallest possible value for f(4)?

- July 27th 2006, 12:25 PMCaptainBlackQuote:

Originally Posted by**Nimmy**

f(1)+3*2=9

RonL - July 27th 2006, 12:57 PMThePerfectHackerQuote:

Originally Posted by**Nimmy**

Honestly, I do not understand what Captain**Blank**is doing.

You are going to use the most impostant theorem in calculus,*mean-value theorem*.

To remind you.

If a function is countinous on and diffrenciable on then there exists are number such as,

-----

Now comes the answer.

Since for all you know that the function is diffrenciable for any (because it exists between 2 and 3). Furthermore, the function is countinous for any because diffrenciability implies counintuity.

Consider on the interval . By the previous explanation we have shown this function satisfies the conditions of the mean value theorem. Thus there is a such as,

. Thus,

Thus,

Thus,

Thus,

Therefore the largest possible value of is the largest possible value of which is 3. Thus,

And the smallest possible values is similarly,

- July 27th 2006, 01:44 PMtopsquarkQuote:

Originally Posted by**ThePerfectHacker**

**Black**'s method was much simpler. I had to read yours twice to see what you were doing. ;)

-Dan - July 27th 2006, 01:59 PMThePerfectHackerQuote:

Originally Posted by**topsquark**

- July 27th 2006, 02:06 PMtopsquarkQuote:

Originally Posted by**ThePerfectHacker**

You basically said the same thing, just with a lot more precision.

-Dan

Tsk, tsk! You aren't supposed to edit quotes! :p - July 27th 2006, 07:47 PMSoroban
Hello, Nimmy!

Quote:

Given that and for all ,

what is the smallest possible value for ?

This is what the Captain meant . . .Code:`| Q`

| * (4,?)

| :

| :

| P :

| * :

| (1,3) :

| :

| :

- - + - + - + - + - + - -

| 1 2 3 4

We are given point

When , we have a point

means:

. . the slope of the graph from to is between and .

The*lowest*occurs when the slope is**exactly 2**(all the way).

This puts at .

Therefore, the minimum value of is

- July 27th 2006, 08:15 PMCaptainBlackQuote:

Originally Posted by**topsquark**

explanation the answer is obvious.

Since symbolic manipulation is error prone, what the answer emerges at

the end of a long chain of symbolic games leaves considerable doubt as

to the correctness of the argument without considerable checking.

Not that the rigorous derivation of the result is not needed some times

to confirm the obvious.

I have an amusing anecdote about this sort of thing, but I won't repeat it

here as I suspect I'm the only one who will find it amusing.

RonL