Ytukyg

Identify the graphs $A$ (blue), $B$( red) and $C$ (green) as the graphs of a function and its derivatives:

1. _____ is the graph of the function


2. _____ is the graph of the function's first derivative


3. _____is the graph of the function's second derivative

1.) A

2.) B

3.) C

The red graph is zero when the blue graph reaches it's local extrema, and the green graph is zero when the red graph reaches it's local extrema. As the other similar relations do not hold the result follows.

Edit: Notice that the blue graph is similar to the graph of $frac{sin^2(x)}{x^2}$.

the function has the blue graph.

the first derivative is zero when the function reaches an extremum, its graph is the red one.

the second derivative gives information on curvature. It is positive when the function decreases and increases just after.
it is negative when the function increases and then decreases.
its graph is the green one.

Here's a list to follow;

1. Check local minimum and maximum. These are going to be zero values in the first derivative since their tangent is parallel to x axis.


2. Check if the graph's slope is increasing or decreasing in a specific point. If increasing the derivative will be in positive side of the y-axis.


3. Look at the sign changes of the first derivative in order to find zero's of the second derivative.

Following the list, the answer will be;
f(x)=A
f'(x)=B
f''(x)=C