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~Finding the largest or smallest value of a quantity is one of the most important type of problem one deals with in calculus. These are referred to as optimization problems.
~There is a wide range of applications from business and science that deal with these types. In business, we want profit to be the largest while cost of manufacturing to be a minimum. In science, we might want to maximize the strength of an object while minimizing its size. There are an enormous number of problems that deal with these concepts.
~These problems can be dealt with effectively if we are able to model the problem with mathematical formulation. Sometimes, it's a very good model of the problem and sometimes not. Most real life situations are too complex to get exact models. Modeling electrical circuits & their behavior comes very close to exact.
~Let me start with definitions:
1) An absolute maximum for f(x) (also called a global max) at x=a
occurs if and only if f(a)≥f(x) for all x in the domain of f.
(i.e., there is no value for f(x) greater than f(a))
2) An absolute minimum for f(x) (also called a global min) at x=c
occurs if and only if f(c)≤f(x) for all x in the domain of f.
(i.e., there is no value for f(x) smaller than f(c))
3) A relative maximum for f(x) (also called a local max) at x=d
occurs if and only if f(d)≥f(x) for all x near d (in a neighborhood of
x=d). A neighborhood is an open interval containing the point.
4) A relative minimum for f(x) (also called a local min) at x=k
occurs if and only if f(k)≤f(x) for all x in some neighborhood of
point k.
~Note: If f is continuous on a closed interval, [a;b], then f will attain an absolute max and absolute min over [a;b]. This is known as the Extreme Value Theorem.
~Note: Not true if we have an open interval (a;b). Take, for example, f(x)=x2 over (0;1).
There are no biggest or smallest values for f(x). Think about it. If you think there are, let me know.
~Note: So, just because a function is bounded (see GLB & LUB link), it does not mean it will have extreme values. Other bounded functions, with horizontal asymptotes, also fall into this category.
~Note: If f has an extreme value at c and if f ' (c) exists, then f ' (c)=0.
(i.e., this will give you one place to look for the max or min).
~Are there other places where we can have extreme values?
~Yes, these are the possibilities (called Critical points)
1) Repeating the above, where f ' (x)=0
2) Endpoints of closed intervals on the x axis. (i.e., if f is defined over
the closed interval, [a;b], we need to check x=a and x=b for
extreme values).
3) Where f ' (x) does not exist. It is possible for a function to have an
extreme value at a cusp.
~Note: Many math teachers do not call extreme values at the end-pts relative or local max or min, they refer to them as an endpoint max/min.
I do, since it does satisfiy the definition of a relative extrema. There still exists neighborhoods of points at the ends of a closed interval, even if the x's come from one side only.
~Time for a simple example (more involved examples will be given in class).
~Ex: Find the absolute maximum and minimum values for f(x)=x2/3 over the closed interval [-1;8].
(we are guaranteed the existence of these values by the Extreme Value Thm)
First, compute f '(x) =(2/3)x-1/3. We need to find the critical points.
1) f '(x) = 2/(3x1/3) can never equal zero, so no values come from here.
2) at x= -1, f(-1) = 1, and at x=8, f(8) = 4. All we need to do is compare these to the values we get in the next step for our answers.
3) f ' (x) fails to exist at x=0. At x=0, f(0) = 0 . There is a cusp at x=0 with a vertical tangent line.
So, just compare the values of the function at the critical values found & we should have our answers.
Therefore, the absolute max = 4 occurring at x = 8.
& the absolute min = 0 occurring at x = 0.
Sketch this curve in your graphing calculator & see for yourself.
~Note: Applied problems are much more interesting & I will cover a variety of types in class.