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### Maximum Points and the Second Derivative

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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.
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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/
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Associated Topics:
High School Calculus

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