Arranging Numbers on a Derivatives Graph: Explanation and Help

[neik]

Help me please!

For the function f whose graph is given, arrange the following numbers in increasing order and explain why:

• 0
• f'(2)
• f(3) - f(2)
• 1/2[f(4) - f(2)]

 

[UrbanXrisis]

0 is undefined...on an asymptote

out of f'(2), f(3) - f(2) and 1/2[f(4) - f(2)]

increasing order:

1/2[f(4) - f(2)]
f(3) - f(2)
f'(2)

 

[neik]

but why?

 

[vsage]

a. 0
b. f'(2)
c. f(3) - f(2)
d. 1/2[f(4) - f(2)]

0 is the smallest and I'll explain why in a second.
f(3) - f(2) can be rewritten as (f(3) - f(2)) / (3-2). This slope is only a tiny fraction smaller than b because if you notice the slope is gradually decreasing as x gets larger. 1/2(f(4)-f(2)) can be rewritten as (f(4)-f(2))/(4-2). This looks like the average slope over that interval which you will notice was slightly less than c (which is in turn slightly less than b)

b, c, and d are positive though because the graph increases from 0-infinity apparently so a is the smallest. Ok I pretty much did all the work for you whoops.