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)?

Question

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)?

amistre64 · Answer

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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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

amistre64 · Answer

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

anonymous · Answer

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 · Answer

Sorry, @polpak - how do we know that?

amistre64 · Answer

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

anonymous · Answer

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

anonymous · Answer

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

myininaya · Answer

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

amistre64 · Answer

thats what I thought :)

amistre64 · Answer

but with a 12 lol

anonymous · Answer

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

myininaya · Answer

yay thats what i got except differently

myininaya · Answer

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

anonymous · Answer

Does that make sense Chap?

anonymous · Answer

How do you figure what f(14) is?

myininaya · Answer

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

myininaya · Answer

I used the mean value thm just like you said

anonymous · Answer

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

anonymous · Answer

Oops wait

anonymous · Answer

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

myininaya · Answer

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

anonymous · Answer

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

anonymous · Answer

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

myininaya · Answer

how do you solve x/5=4

anonymous · Answer

$$-4a < b \implies a \gt \frac{b}{-4}$$

anonymous · Answer

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

anonymous · Answer

Multiply by -1

myininaya · Answer

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

anonymous · Answer

135 <= f(14)

myininaya · Answer

right!

myininaya · Answer

wait where's the negative?

anonymous · Answer

That's the answer?

anonymous · Answer

-135 <= f(14)

myininaya · Answer

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

anonymous · Answer

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

anonymous · Answer

It might be positive!

anonymous · Answer

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

myininaya · Answer

it could be 1111111111111111111111

anonymous · Answer

over 9000!

myininaya · Answer

5 trillion billion if that means anything