Could someone please help me on this question,

Suppose f(0) = 2 & -2

Question

Could someone please help me on this question,

Suppose f(0) = 2 & -2<f'(x)<1 for -5<x<5

Find an upper bound for f(4) = (answer = f(4)<6)

jamesj · Answer

Nice.

Well, the derivative tells you how fast f can grow.  So f'(x) < 1 tells you f can grow no faster than f'(x) = 1.  I.e., at a rate of x

So whatever else f is or f(4) is, we know that

$$\frac{f(4) - f(0)}{4 - 0} < 1$$

Can you see now how to manipulate this to get an upper bound on f(4)?

anonymous · Answer

thanks for the reply, though where would we get the 4 in the first place?

anonymous · Answer

Oh sorry I get it haha, its asking that lol, so to manipulate it, you would um

anonymous · Answer

Honestly, I kind of confused on how to go further from that,

anonymous · Answer

you can think of it this way if you like
the derivative being bounded above by one is like the speed limit.  you can walk at most 1 mile per hour.  so if at time 0 you are at mile marker 2, what is the furthest you be in 2 hours?

anonymous · Answer

excuse me i mean what is the furthest you can be in 4 hours?

jamesj · Answer

Nice analogy.

anonymous · Answer

thnx.  too bad i messed up and said 2 hours instead of 4

anonymous · Answer

Oh ok thanks for that, so then since we are at f(4), and the fastest we can move is 1, so how would I get the 6 out of that, sorry its the first time i tackle upper and lower bounds :@

anonymous · Answer

Would I graph it?

anonymous · Answer

Oh i got it, thanks a lot for the help mates, I appreciate it.