Here's the question you clicked on:
swin2013
For the function graphed, are the following nonzero quantities positive or negative?
|dw:1354148967410:dw| a) f(2) - positive b) f'(2) - ? c) f''(2) - ?
f'(2) represents the SLOPE of the function at that point. So let's draw the line tangent to the curve at f(2) to see what it looks like.
|dw:1354149736096:dw| It's kind of hard to tell, based on the sloppy drawing lolol. Was it supppose to be slanted down like that? Or is it suppose to be flat on the bottom at 2?
lol it's suppose to look like a the tip of a parabola
So a horizontal line represents what kind of slope? :D
f'(2) is zero :) we can tell that since we drew a tangent line. It is neither positive nor negative, right? :D
so neither are postive or negative?
except for f(2)? Which is negative.. lol i never meant to put postive
f'' is a little trickier. There are a couple ways we can determine if it's positive or negative. f'' tells us if the SLOPE is increasing or decreasing. But an easier way to maybe think about it is by referring back to the second derivative test. \[f''>0 \qquad \rightarrow \qquad \text{Concave Up}\]\[f''<0 \qquad \rightarrow \qquad \text{Concave Down}\]
We can use this information in reverse. In the given problem we KNOW that f(2) lies in an area that is CONCAVE UP. See how it's inside of a bowl shape? So what does that tell us about f''?
Well... it tells us that f' is increasing c: But based on the little facts I printed a sec ago, it tells us that f'' is GREATER THAN ZERO right? :D or in other words, Positive.
so f(2) is negative, f'(2) is positive, therefore f''(2) is also positive :D
Why was f'(2) positive? :o
We determined that the slope is zero. It will increase towards the positive as we move to the right, but it is zero at the point f(2).
f(2) Negative f'(2) Zero f''(2) Positive
ohhh... it's because they asked if they're positive or negative lolll
Yah, these kinds of uhhh graphs dealing with derivatives can be quite tricky c:
Lolll ahhh well thank you again!! you're a life saverrrr!