Find Global Extrema of Functions: Steps & Example

In summary, finding global extrema of functions involves four steps: 1) Finding critical points, 2) Finding values of the function at critical points and end-points, 3) Identifying the largest and smallest values, and 4) Determining the global maxima and minima. To find local extrema at end-points, it is helpful to remember that an end-point is always a local extremum. If the function is increasing, the end-point is either a max or min depending on the interval, and if the function is decreasing, the end-point is reversed.
  • #1
Ali Asadullah
99
0
Please check my steps for global extrema of functions.
1) Finding Critical points.
2) Finding values of f at critical points and end-points.
3) Checking which one is largest and which is smallest.
4) Largest would be Global maxima and the smallest would be minima.

Now a question, i know how to find global extrema if they are at
end-points. How can i find local extrema if they occur at end-points?
I will be thankful to you if you illustrate it with a simple example.
 
Physics news on Phys.org
  • #2
Strictly speaking, and endpoint is always a "local extremum". If the function is increasing there, then the endpoint is either a max or a min depending on which side the interval is. If the function is decreasing, then that is reversed.
 

What is the definition of global extrema?

Global extrema are the highest and lowest values that a function can take on within a given interval or domain. They represent the absolute maximum and minimum values of a function.

How do you find the global extrema of a function?

To find the global extrema of a function, you must first determine the critical points by taking the derivative of the function and setting it equal to 0. Then, evaluate the function at each critical point and at the endpoints of the interval to determine the highest and lowest values.

What is the difference between global and local extrema?

Global extrema are the highest and lowest values of a function within a given interval, while local extrema are the highest and lowest values within a small region or neighborhood of a point on the function. Global extrema are absolute, while local extrema can change depending on the chosen region.

Can a function have multiple global extrema?

Yes, a function can have multiple global extrema if the interval is large enough and the function has multiple peaks and valleys. In some cases, a function may have infinitely many global extrema.

What is the significance of finding global extrema?

Finding global extrema allows us to determine the overall behavior and characteristics of a function. It can help us identify maximum and minimum values, as well as the overall trend of the function. This information is useful in many fields, including economics, physics, and engineering.

Similar threads

MHB Find where increasing/decreasing, concavity, local extrema and inflection points for f(x)=ln/x
    • Calculus
Replies
1
Views
1K
MHB Extrema of Functions of Two Variables
    • Calculus
Replies
4
Views
2K
MHB How to Find the Absolute Extrema of a Function Over a Defined Region?
    • Calculus
Replies
2
Views
1K
MHB Are There Global Extrema for These Functions?
    • Calculus
Replies
10
Views
3K
MHB Finding Critical Points and Extrema for g(x, y) = sqrt{x^2 + y^2 + 1}
    • Calculus
Replies
1
Views
1K
Confusion with Relative Extrema and Intervals
    • Calculus
Replies
7
Views
2K
MHB Find Absolute Extrema Over Region R
    • Calculus
Replies
3
Views
4K
MHB Critical Points & Extrema of f (x, y)
    • Calculus
Replies
1
Views
1K
MHB Critical Points & Extrema of Multivariable Function
    • Calculus
Replies
1
Views
1K
MHB Do we get the extrema from that graph?
    • Calculus
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top