December 3rd, 2017, 03:49 PM  #1 
Senior Member Joined: Apr 2017 From: New York Posts: 155 Thanks: 6  derivative of the function
hi guys, I am trying to match the function with its derivative on the graph. Any idea which could be the derivative of the blue line and why? thanks a lot. 
December 3rd, 2017, 03:56 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,949 Thanks: 1141 Math Focus: Elementary mathematics and beyond 
The derivative is zero where the function has a local min/max. The steeper the curve is, the greater in value the derivative is  with negative "steepnes" having a greater negative value and positive steepness having a greater positive value. In short, the greater the absolute value of the derivative the more steeply the curve is inclined. Does that help?

December 3rd, 2017, 04:34 PM  #3 
Senior Member Joined: Apr 2017 From: New York Posts: 155 Thanks: 6 
so which one is the answer below, please?

December 3rd, 2017, 05:04 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,949 Thanks: 1141 Math Focus: Elementary mathematics and beyond 
It's hard to tell without knowing where the xaxis is. Besides, I was hoping you'd be able to answer that. What do you think?

December 3rd, 2017, 05:19 PM  #5 
Senior Member Joined: Apr 2017 From: New York Posts: 155 Thanks: 6 
I eliminated number 1 and 2 on my mind but this is a sensitive question that I have only one chance to input. not a second chance is given so wanted to make sure 100%. According to your explanations it seems 3. but again need to make sure before inputting. 
December 3rd, 2017, 05:37 PM  #6 
Math Team Joined: Jul 2011 From: Texas Posts: 2,975 Thanks: 1571  Not 3 ... note the slope of the original function is negative, positive, and negative from left to right. Also note the slope is positive at x = 0 
December 3rd, 2017, 06:02 PM  #7 
Senior Member Joined: Sep 2016 From: USA Posts: 634 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics 
It is #4. Now, with that known, can you figure out why? What does the derivative tell you about the function? Hint: Look at the 2nd derivative. 
December 3rd, 2017, 11:04 PM  #8 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,949 Thanks: 1141 Math Focus: Elementary mathematics and beyond  
December 4th, 2017, 11:15 AM  #9 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
One of the first things you should have seen from the original graph (of the derivative) is that the derivative is positive from some negative x value to some positive x value. There must be an interval from a negative x to a positive x where the graph is positive. Only one of the four graphs has that property.

December 4th, 2017, 02:24 PM  #10 
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150 
As the curve crosses the vertical line, its slope (derivative) changes from increasing to decreasing, so it has a local maximum at the point where it crosses the vertical line. Only the "last" graph has that property.


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