The maximum of a function is found by first computing the derivative of the function
, as it describes the rate of change of the original function
. At a maximum, the slope of the function changes from positive where the function is increasing to negative where the function is decreasing. Therefore, the slope is equal to zero at a maximum:
. The points at which
are denoted as critical points. It is important to note that not all critical points are maximums since there are other instances where the slope is equal to zero, such as a minimum or inflection point of the function. To determine whether a critical point is a maximum, the second derivative of the function
, must be computed. It describes the concavity of the function and how its slope is changing. If
, the function is concave down and the critical point is a maximum.
Find the maximum of the function
Determining the critical points
Compute the derivative
of the function.
Since
takes the form
, it can be differentiated using the product rule.
Product Rule
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![{\displaystyle {\frac {d}{dr}}[r^{2}\exp(-r^{2})]={\frac {d}{dr}}(r^{2})\exp(-r^{2})+(r^{2}){\frac {d}{dr}}\exp(-r^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b07cea826d89c047c52007a60ffdb7da383e8f05)
Since a maximum occurs when the slope is zero,
The points at which the function equal zero are the critical points. Since
will never equal zero,
when
when
Therefore, the critical points are
Evaluating the critical points
Compute the second derivative
of the function.
Since
again takes the form
,
Evaluating
at the critical points determines the concavity of the function.
When
, the critical point is a maximum.
At
,
.
At
,
At
,
Since
, the points
are maximums.