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Maximum Points and the Second Derivative


Date: 04/24/98 at 05:13:11
From: Sue
Subject: Second Derivative

Why is that when the second derivative of a point is a negative, the 
point is said to be a maximum? I'm a teacher and I just state that 
fact and never explain why.


Date: 04/24/98 at 12:19:34
From: Doctor Sam
Subject: Re: Second Derivative

Sue,

In general it is not true that when the second derivative of a 
function at a point is negative that the function has a maximum. For 
example, the function y = 2 - x^2 has first derivative y' = -2x and 
the second derivative is y'' = -2. This is negative at every point but 
the function only has a maximum at x = 0.

It is true that if the second derivative is negative "at a point where 
the first derivative is zero" that point will be a maximum value of 
the function. Here's why.

The first derivative of a function at a point gives the slope of a 
line tangent to the graph of the function at that point.  So if 
y = 2-x^2 (so that y' = -2x) then the slope of the tangent to this 
parabola at x = 3 has slope -6.  

In general, whenever the first derivative is negative, the graph is 
sloping down at that point and so the function is decreasing. Whenever 
the first derivative is positive at a point, the function is 
increasing.

A relative maximum value of a function is the highest point among all 
the nearby points. (If you are walking in a hilly region you can be 
standing on top of a hill, and so be higher than nearby points, and 
yet not be standing at the highest hill.) If you imagine walking along 
a graph with your x-values increasing, then you must walk up the graph 
until you reach a relative maximum, and then you must walk down the 
graph as you leave it. That is, the slope of the graph as you approach 
a maximum must be positive and the slope as you leave the high point 
must be negative. This gives one way to find the maximum values of a 
function: look for points where the first derivative changes sign from 
positive to negative. The value of the first derivative at these 
points is often zero, but sometimes the graph might have a sharp 
corner (like y = |x| has at x = 0). Here the derivative does not 
exist. There is no tangent line possible to the absolute value 
function at x = 0. But the slopes of y = |x| change at that point from 
negative to positive.  

Points where the first derivative is zero or does not exist are called 
"critical points" of the function, because they are critical to the 
analysis of where the function has a maximum or a minimum.

But you are asking about the second derivative. If you think of the 
second derivative as "the derivative of the first derivative" then a 
negative second derivative means that the slope of the derivative 
graph is negative - that is, the derivative is decreasing. If the 
first derivative is decreasing, that means that the slopes of the 
original function are decreasing. If you try sketching a curve whose 
slopes are getting smaller and smaller, you will discover that you 
must draw a curve that is concave down (looking like an inverted 
bowl). Likewise, if the second derivative is positive, then the slopes 
of the function are increasing and so the original function is concave 
up (like a U).  

So the second derivative tells us about the concavity of a function.  
Finally, here is the reason you seek: 

  if a function's second derivative is negative at a critical point,
  then the function has a relative maximum there. 

The reason is that the graph is concave down at a point where the 
graph levels off ... like the top of a hill.  This is called the 
second derivative test. The other half of this test is:

  if the function's second derivative is positive at a critical
  point then the function has a relative minimum there.

This is because the graph is concave up at a critical point, which is 
what happens in shapes like y = x^2.

I hope that helps.

-Doctor Sam,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus

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