Suppose that we know that f(x) is a continuous and differentiable function on [2,14]. Also, suppose we know that f(2)=9 and f'(x) is greater than or equal to -12. What is the smallest possible value of f(14)?

- anonymous

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- amistre64

would this be a straight line from 2,14 running at a slope of 12?

- anonymous

(I'm supposed to use the mean value theorem, or at least that's what it says on the worksheet.)

- anonymous

(I'm supposed to use the mean value theorem, or at least that's what it says on the worksheet.)

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## More answers

- anonymous

@amistre64 - I have no idea! I'm complete lost on this one. Don't even know where to start.

- amistre64

f'(2) =>12....6x? maybe

- anonymous

Well, if f'(x) = -12 throughout the interval then it would be a straight line from (2,9) with a slope of -12. So that would be the minimum possible value given what we know.

- anonymous

Sorry, @polpak - how do we know that?

- amistre64

ahh,....(-)12 ... that was hiding in the corner

- anonymous

We are given that f'(x) is greater than or equal to -12.

- anonymous

So if it maintained the smallest possible derivative over the whole interval we would have a line with slope of -12.

- myininaya

i got the smallest possible value is -135 for f(14)

- amistre64

thats what I thought :)

- amistre64

but with a 12 lol

- anonymous

9-12*12 = 9-144 = -135

- myininaya

yay thats what i got except differently

- myininaya

-12<=[f(2)-f(14)]/[2-14]

- anonymous

Does that make sense Chap?

- anonymous

How do you figure what f(14) is?

- myininaya

chap you can also solve the above for f(14) resulting in f(14)>=(-135)

- myininaya

I used the mean value thm just like you said

- anonymous

Right. But I have -12<= 9 - f(14)

- anonymous

Oops wait

- anonymous

I have -12 <= 9 - f(14)
-----------
-12

- myininaya

multipliy -12 on both sides don't forget to flip the inequality when you multipliy or divide by a negative

- anonymous

Sorry )= I don't understand how to do that.

- anonymous

\[-12 \le \frac{9- f(14)}{-12} \implies 144 \ge 9-f(14)\]

- myininaya

how do you solve x/5=4

- anonymous

\[-4a < b \implies a \gt \frac{b}{-4}\]

- anonymous

Okay. But how does 135 >= - f(14) solve for f(14)?

- anonymous

Multiply by -1

- myininaya

now multipliy both sides by negative 1 don't forget to flip

- anonymous

135 <= f(14)

- myininaya

right!

- myininaya

wait where's the negative?

- anonymous

That's the answer?

- anonymous

-135 <= f(14)

- myininaya

ok so we have f(14)>=-135 this means the smallest that f(14) can be is -135

- anonymous

OH! I thought f(14) would be positive. Thank you so much!

- anonymous

It might be positive!

- anonymous

We only know that the smallest it can be is -135. It could be very very large.

- myininaya

it could be 1111111111111111111111

- anonymous

over 9000!

- myininaya

5 trillion billion if that means anything

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