True Maximum and Minimum Values
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In the last section, we found methods to get the relative minimum and
maximum values of a function. Now, suppose that we want to find the highest
and the lowest values of a function in a closed interval I. These values are not necessarily relative maximum
and minimum values of the function. Consider the case of |
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NB |
A true extreme value in a closed interval is either at an endpoint of
the interval or a relative extreme value. Therefore, to find the true maximum
and minimum values of a function, we simply do the following steps: 1- We find the critical points of the function at the given interval (Of course that will include all relative extreme points.), and we find the values of the function there. 2- We find the values of the function at the
endpoints of the interval. 3- The largest value we get from these two steps is
the true maximum value at the interval. The lowest value is the true minimum
value. |
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NB |
We stress that we search for extreme values inside a closed interval. The
function may not have a true extreme value in an open interval, such as the |
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Example |
An open box in the form of a cuboid is to be made from a rectangular
piece of cardboard of length 16cm and of width 10cm by cutting four equal
squares of each of x centimeters sides from the corners of the rectangle and folding the
protruding parts to form the box. Find the value of x for which the volume of the
box is a maximum. |
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. There are relative extreme values at 
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