Question

Examine the graphs below and determine which is the graph of f(x), f ′(x), f ′′(x):

Answer 1

I marked the answer that I think is correct, but, even if I did get it right, I don't understand fully how I'm supposed to know the difference between the graphs?

Answer 2

Well you have a few options :)

You can try to read the slopes at difference places and connect the information that way (which is probably a good way to solidify some of these calculus techniques)

or you can recognize that they've graphed polynomial functions.
One of them appears to be a 4th degree polynomial,
another one is a 3rd degree,
and the last one looks like a simple parabola, ya?

Answer 3

derivative of a 4th degree polynomial gives you a 3rd degree polynomial, right?
They decrease in power by 1 each time you differentiate.

Answer 4

So, since they decrease in power, would it be the third option? And, in that case, do I always start from the highest degree? Because that kind of makes sense, if so

Answer 5

Well I should be careful in saying that A "looks like" a simple parabola.
It's possible that it's a 6th degree polynomial that doesn't have a bunch of turns.
But yes.

Answer 6

The shape just tells us that it's "even"

Answer 7

hold on lemme draw it a sec XD
understanding the slope would be really helpful

Answer 8

it is not possible that it is a 6th degree polly

Answer 9

Let's pretend that A is f(x), and see how that would work out.

Answer 10

Notice that the line tangent to A is zero at x=0.