Max and min in several variables

In summary, the maximum value of f(x,y)= x^2-2x+y^2 for f(x,y) in the region bounded by the triangle with vertices (1,0) (1,1) (0,1) is 1, and the minimum value is -1. The method used to find these values is by finding the critical points and evaluating the function at these points, as well as the boundaries of the region.
  • #1
Derill03
63
0
Find max value and min value of f(x,y)= x^2-2x+y^2 for f(x,y) in the region bounded by the triangle with vertices (1,0) (1,1) (0,1)

Im not sure if i did this right, i used second derivative test to find a min at -1, and by plugging in (0,1) i found the max to be 1. But i can only get max value by plugging in (0,1) is there another way to find max that i am missing? And is the way i did this correct?
 
Physics news on Phys.org
  • #2
Ummm...

Well, first we look for relative extrema. Setting

2x - 2 = 0
2y = 0

We get x = 1, y = 0. So (1, 0) is the only critical point.

We must now check the value of the function at the critical point, as well as for points on the triangular boundary. So

f(1,0) = -1.

The first boundary is (1,0) - (1,1) which means x = 1, y between 0 and 1. So
f(1, y) = y - 1 which is maximum at y = 1, so this boundary has a max of 0 at (1,1).

The second boundary is (1,1) - (0,1), which means y = 1, x between 0 and 1. So
f(x,1) = x^2-2x+1 = (x-1)^2 which is maximum at x = 0, so this boundary has a max of 1 at (0, 1).

The third boundary is (0,1) - (1,0), which means y = 1-x, x between 0 and 1. So
f(x,1-x) = x^2-2x+(1-x)^2 = x^2-2x+1 -2x+x^2 = 2x^2-4x+1 which is maximum at x = 0, so this boundary has a max of 1 at (0, 1).

So the candidates are (1,0)=>-1, (1,1)=>0, (0,1)=>1, and (0,1)=>1.

Clearly, (0,1)=>1 is the maximum in the region.
 

Related to Max and min in several variables

1. What is the definition of a maximum and minimum in several variables?

In mathematics, a maximum refers to the largest value in a set or function, while a minimum refers to the smallest value. In several variables, this means finding the largest or smallest value of a function that depends on multiple variables.

2. How do you find the maximum and minimum of a function in several variables?

To find the maximum and minimum of a function in several variables, you must use multivariable calculus techniques such as partial derivatives and the gradient vector. These methods involve setting up and solving equations to find the critical points of the function, and then evaluating the function at these critical points to determine the maximum and minimum values.

3. What is the significance of the maximum and minimum in several variables?

The maximum and minimum values of a function in several variables are important because they represent the extreme points of the function. They can provide information about the behavior and characteristics of the function, and are often used in optimization problems to find the best possible outcome.

4. Can a function have multiple maximum or minimum values in several variables?

Yes, a function can have multiple maximum or minimum values in several variables. These points are called local maxima or minima and can occur when the function has multiple critical points or when the function has a saddle point (a point where the function is neither increasing nor decreasing).

5. How are the maximum and minimum values affected by changing the variables in a function?

The maximum and minimum values of a function in several variables can change when the variables are changed. This is because the function itself may change as the variables are varied. In some cases, the maximum and minimum values may remain the same, while in others they may shift to different points. This is why it is important to consider the variables when finding the maximum and minimum of a function.

Similar threads

MHB 48.2.2.23 IVP and min valuew
    • Calculus
Replies
3
Views
760
I Understanding a proof of inexistence of max nor min
    • Calculus
Replies
4
Views
911
MHB Find a,b,c,d given max and min in cubic function
    • Calculus
Replies
4
Views
1K
MHB Find where increasing/decreasing, concavity, local extrema and inflection points for f(x)=ln/x
    • Calculus
Replies
1
Views
1K
MHB Absolute max/min over intervals
    • Calculus
Replies
3
Views
2K
MHB Solve Optimization Problem: Find Min & Max of f(x,y)
    • Calculus
Replies
3
Views
2K
MHB How Do You Find the Absolute Min and Max of a Function on a Triangular Region?
    • Calculus
Replies
4
Views
2K
I Lipschitz continuity of vector-valued function
    • Calculus
Replies
3
Views
1K
I Function of 2 variables, max/min test, D=0 and linear dependence
    • Calculus
Replies
7
Views
1K
Min/max of functions restricted to subsets of their domains
    • Calculus
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top