Here’s another chapter of material involving more ways to apply the derivative to several other types of
problems. This stuff focuses mainly on using the derivative to aid in graphing a function, etc.
APPLIED MAXIMA AND MINIMA PROBLEMS
One of the most common applications of the derivative is to find a maximum or minimum value of a
function. These values can be called extreme values, optimal values, or critical points. Each of these
problems involves the same, very simple principle.
A maximum or a minimum of a function occurs at a point where the derivative of a function is zero,
or where the derivative fails to exist.At a point where the first derivative equals zero, the curve has a horizontal tangent line, at which point it
could be reaching either a “peak” (maximum) or a “valley” (minimum).
There are a few exceptions to every rule. This rule is no different.
If the derivative of a function is zero at a certain point, it is usually a maximum or minimum—but not
always.There are two different kinds of maxima and minima: relative and absolute. A relative or local
maximum or minimum means that the curve has a horizontal tangent line at that point, but it is not the
highest or lowest value that the function attains. In the figure to the right, the two indicated points are
relative maxima/minima.