How to solve the problem:
There are two indicators that one should look for when identifying the graphs of higher derivatives — both indicators are the result of recognzing certain facts about derivatives. The first indicator is to recognize that the points at which the the tangent line is horizontal. These horizontal tangent lines occur at minima and maxima of graphs (the peaks and troughs of a graph). It is extremely improtant to remember that when the tangent line is horizontal the derivative is equal to zero. The second indicator is whehter a particular interval on the graph is increasing or decreasing. If an interval along the graph is increasing the derivative has a positive value, while if an interval is decreasing the derivative has a negative value. The graph of the derivative along the interval of the function that is decreasing should be below the x-axis, i.e, negative. In contrast the graph of the derivatice along the interval of the function that is increasing should above the x-axis, i.e, positive . Putting these together, one could determine what graphs are derivatives of others.
In the problem given notice that the graph of function A has a maximum that is right on the y-axis i.e, the tangent line across this maximum is horizontal. In other words, at this point the derivative is equal to zero. Therefore the graph of the derivative of the function of A should be crossing the x-axis at pricisely the same x-value that the maximum is located. And indeed, one notices that the graph of the function B obeys this behavior: It crosses the x-axis at pricisely the same x-x-axis value that the maximum for the graph of A is located. Evaluating the extrema for the graph of B should drive this point home and display the ease at each the graphs could be determined. Notice that the graph of B has maximum and and minimum. At both these points, the tangent line horizontal, again, indicating that the derivative is zero. Therefore, the graph of the derivative of B should have two points at which it crosses the x-axis at pricisely the same x-value that extrema (maximum or minimum) is located. If one looks carefully it is clearly evident that the graph of C obeys these factors. These observation alone should be able to convince anyone that graph B is the derivatice of A and graph C is the derivative of the graph of C: A=f, B=f’, and C= f”. However, for more complex graphs analyzing increasing and decreasing intervals could be necessary.
At any rate, this my analysis (it may be a little confusing and not very clear). Any feedback and questions are welcomed.
-Abdoulaye
