Question

Hey everyone,

I'm really confused on the concept of upper and lower bounds. An example would be:

Suppose that f(3)=4 and 2(equalto or <) f'(x) (equalto or <) 3 for all x.

Find an upper bound on the value of f(5).

Please explain with steps so I can learn :)

Thank you.

Answer 1

ok we can thing of it like this. think that
\[f'(x)\] is the speed limit. this says your speed is between 2 and 3.
now at 3 pm you are at mile marker 4 on the highway. when can you be at 5 pm?

Answer 2

5 pm is 2 hours later, so at most you could have gone
\[2\times 3=6\] miles and now you are at mile marker 10 (since 6+4=10)

Answer 3

or you could have traveled a minimum of
\[2\times 2=4\] miles so if you went the minimum speed you would be at mile marker 8 (because 4+4=8)

Answer 4

meaning
\[8\leq f(5)\leq 10\]

Answer 5

we can write this using the mean value theorem if you like.

Answer 6

\[2\leq \frac{f(5)-f(3)}{5-3}\leq 3\]

\[2\leq \frac{f(5)-4}{2}\leq 3\]
\[4\leq f(5)-4\leq 6\]
\[8\leq f(5)\leq 10\]

Answer 7

Oh I get it, thanks a lot for you help!

Answer 8

yw