Question

Suppose that 2 ≤ f '(x) ≤ 5 for all values of x. What are the minimum and maximum possible values of f(6) − f(3)?

ATTEMPT: Unfortunately for this one, I am not sure how I am going to proceed with it. If anyone cares to help or give a hint, that would be great.

The answer comes in blanks like this:
[ ] ≤ f(6) − f(3) ≤ [ ]
Suppose that 2 ≤ f '(x) ≤ 5 for all values of x. What are the minimum and maximum possible values of f(6) − f(3)?

ATTEMPT: Unfortunately for this one, I am not sure how I am going to proceed with it. If anyone cares to help or give a hint, that would be great.

The answer comes in blanks like this:
[ ] ≤ f(6) − f(3) ≤ [ ]
@Mathematics

Answer 1

Wow how did you get to this page so fast

Answer 2

the calculus call to me

Answer 3

HAHA

Answer 4

It would be 0\[0\le f(6)-f(3) \le 3\] I believe

Answer 5

Notice that f is always increasing

Answer 6

Aw shoot. I should've said I didn't want an answer. Its alright though. The reason being is that I don't really understand if I get the answer in front of me. I can get that sensation by looking at the back of the textbook.

Answer 7

Yes f is increasing throughout the entire interval

Answer 8

oh... I'm sorry.

Answer 9

another hint: secant line

Answer 10

Oh I see, so the end values of the interval I need to use them?

Answer 11

wait...but it doesn't say that f(n) increases as n increase. I figured that the smallest difference would be 0, because two n can produce the same f(n) result. And the biggest would be 3, where one is 2 and the other is 3.

Answer 12

ah!!!! I gave you the wrong answer way up there ^, not the previous reply, but the first one that I put.

Answer 13

Wait a second. Can I use antiderivation to solve this?

Answer 14

yes, but if we notice that f' is between 2 and 5 it is increasing in that interval, which denote that the slope is positive, which further means that the orginal function is increasing

Answer 15

So what if the function is increasing?

Answer 16

it help us because now we are able to use the secant line to determine or solve this inequality, since it will have a slope between 2 and 5

Answer 17

Equivalently, by the Fundamental Theorem of Calculus,
\[ f(6) - f(3) = \int_3^6 f'(x) \ dx \]

Now as \( 2 \leq f'(x) \leq 5 \), it follows that
\[ \int_3^6 2 \ dx \leq \int_3^6 f'(x) \ dx \leq \int_3^6 5 \ dx \]

Hence ... I'm sure you've got it from here.

Answer 18

got it?

Answer 19

I'll work on it from here. Thanks for your help.

Answer 20

tell me what final answer you get. It's actually a nice problem.