Local and Global, Maximums and Minimums
Hello class, in today's lesson we are going to be covering how to find local and global maximums and minimums.
Being able to find these values is important because it allows you to find where a function is at its greatest or its smallest. It also allows you to find a value within the function where it changes from decreasing to increasing or from increasing to decreasing.
First lets talk about what these terms mean:
Local: The local maximum or minimum of a function is a point where there is a point with the largest or smallest value relative to the points around it. There can be many local maximums and many local minimums.
Global: The global maximum or minimum means that it is the value of a function that is the absolute largest or smallest. There are no other values that are higher if it is the maximum or lower if it is the minimum. There can only be one global maximum and one global minimum.
Maximum: A maximum or "max" is a value that is the largest or larger compared to other values.
Minimum: A minimum or "min" is a value that is the smallest or smaller compared to other values.
All of these concepts can be seen in this graph.
Now lets talk about how to find these values. When you are given a function that you want to find these values, you can use the First Derivative Test.
The First Derivative Test is a process where you find the derivative of a given function and then set the derivative equal to 0. When the derivative is equal to zero it means that there is a critical point. A critical point can be a local or global, maximum or minimum.
It is a max when the derivative goes from positive to negative.
It is a min when the derivative goes from negative to positive.
It is not a max or min when it stays at negative when it was already negative or stays at positive when it is already positive.
A global max or min is either at the endpoints or a critical point.
Example of First Derivative Test:
On the interval [-2,3]
f(x) = 3x4−4x3−12x2+3
f'(x) = 12x3−12x2−24x
= 12x(x2−x−2)
= 12x(x+1)(x−2)
0 = 12x(x+1)(x−2)
x = -1,0,2
Before -1 the derivative is negative:
12(-2)(-2+1)(-2-2)= -96
After -1 the derivative is positive:
12(-.5)(-.5+1)(-.5-2)= 7.5
This means that at -1 there is a min.
When plugged into the original function -1, gives an output of -2.
This means it is a local min.
Before 0 the derivative is positive:
12(-.5)(-.5+1)(-.5-2)= 7.5
After 0 the derivative is negative:
12(.5)(.5+1)(.5-2)= -13.5
This means that at 0 there is a max.
When plugged into the original function 0, gives an output of 3.
This means it is a local max.
Before 2 the derivative is negative:
12(.5)(.5+1)(.5-2)= -13.5
After 2 the derivative is positive:
12(3)(3+1)(3-2)= 144
This means that at 2 there is a min.
When plugged into the original function 2, gives an output of -29.
This means it is the global min.
At 3 the function gives an output of 30. On the interval this is the global max.
So, x=3 is the global max, x=-1 is a local min, x=0 is a local max, and x=2 is the global min.
That is the end of today's lesson. You now know how to find the local and global, maximum and minimum and what they mean.
Brian, You did a really good job on defining some key terms which help to better understand the rest of the lesson. Also, the graph provided an additional benefit to help clarify what the points look like. Nice job!
ReplyDeletebrian,
ReplyDeletenice job of explaining max's and min's and the concept of the first derivative test. i really liked how you used some good everyday language and mixed it with mathie terms to make the concept easier to digest!
professor little