## anonymous 5 years ago 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)?

1. amistre64

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

2. anonymous

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

3. anonymous

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

4. anonymous

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

5. amistre64

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

6. 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.

7. anonymous

Sorry, @polpak - how do we know that?

8. amistre64

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

9. anonymous

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

10. anonymous

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

11. myininaya

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

12. amistre64

thats what I thought :)

13. amistre64

but with a 12 lol

14. anonymous

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

15. myininaya

yay thats what i got except differently

16. myininaya

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

17. anonymous

Does that make sense Chap?

18. anonymous

How do you figure what f(14) is?

19. myininaya

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

20. myininaya

I used the mean value thm just like you said

21. anonymous

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

22. anonymous

Oops wait

23. anonymous

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

24. myininaya

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

25. anonymous

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

26. anonymous

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

27. myininaya

how do you solve x/5=4

28. anonymous

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

29. anonymous

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

30. anonymous

Multiply by -1

31. myininaya

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

32. anonymous

135 <= f(14)

33. myininaya

right!

34. myininaya

wait where's the negative?

35. anonymous

36. anonymous

-135 <= f(14)

37. myininaya

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

38. anonymous

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

39. anonymous

It might be positive!

40. anonymous

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

41. myininaya

it could be 1111111111111111111111

42. anonymous

over 9000!

43. myininaya

5 trillion billion if that means anything