Thread: Calculus Help Needed!

If f(2)=4 and f'(x)>2 for 2<x<11, how small can f(11) be?

Thanks!

Originally Posted by asnxbbyx113

If f(2)=4 and f'(x)>2 for 2<x<11, how small can f(11) be?

Thanks!

Note that f satisfies the conditions for the mean value theorem to be
applicable on [2,11].

Suppose f(11)<22, then by the mean value theorem there exists a c in (2,11)
such that:

f'(c)=(f(11)-f(2))/9 = (f(11)-4)/9,

but if f(11)<22 this is <2, which contradicts the assumption that f'(x)>=2.

Now let f(x)=4+2(x-2), then f'(x)=2 for x in (2,11), and f(11)=22.

Therefore f(11) can be as small as 22 and no smaller.

RonL

Hello, asnxbbyx113!

Did you make a sketch?

If and for , how small can be?

We consider the graph of from to .

SInce , the graph starts at .

Code:

        | (2,4)
        |   *
        |
        |
      - + - + - - - - - - -
        |   2

From to , the slope is

The very "lowest" graph would be straight line from with slope .
. . Its equation is: .

It would look like this:

Code:

        |
        |                 * (11,22)
        |               * :
        |             *   :
        |           *     :
        |         *       :
        |       *         :
        |     *           :
        |   *             :
        |   :             :
        |   :             :
      - | - + - - - - - - + - -
        |   2            11

Hence, the least value for is: .

Originally Posted by CaptainBlank

Note that f satisfies the conditions for the mean value theorem to be
applicable on [2,11].

I am honored that I taught you something. I remember somebody asked a similar question. You answered it in some other way. I answered it elegantly via MVT. Now it seems you liked mine more.

Originally Posted by ThePerfectHacker

I am honored that I taught you something. I remember somebody asked a similar question. You answered it in some other way. I answered it elegantly via MVT. Now it seems you liked mine more.

Well I've always known about the use of the MVT, but I thought it
inappropriate for the level of knowlege of the student involved, as
it probably is here also, which is why Soroban is thought a better helper
than both of us .

A MVT proof is more appropriate to a first Real Analysis course than to
any course calling itself Calculus X.


It seems some of your tendency to answer questions with methods
above the heads of the asker has rubbed of on me

RonL

Originally Posted by CaptainBlank

.
It seems some of your tendency to answer questions with methods
above the heads of the asker

That is what you think, I do not see how I make myself complicated. I know that I do not try to be simple I consider sme simple approaches mathematically inappropriate and need to use a better one.

Originally Posted by ThePerfectHacker

That is what you think, I do not see how I make myself complicated. I know that I do not try to be simple I consider sme simple approaches mathematically inappropriate and need to use a better one.

If they haven't met the MVT, then such a solution is beyond their experience. Such an answere would have to in effect prove (or otherwise justify the MVT).

RonL