Maximum and Minimum Values Inside Triangle

In summary: Along L1, f can be expressed by the one variable function:f = f(_,y) = 9Along L2, f can be expressed by the one variable function:f = f(x,_) = -5Along L3, f can be expressed by the one variable function:f = f(x,_) = -13
  • #1
Chas3down
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Homework Statement


Find the absolute maximum and minimum values of f(x,y) = y^2+x^2 -4x + 11 on the set D where D is the closed triangular region with vertices (8,0),(0,4) , and (0,-4) .

Homework Equations

The Attempt at a Solution


[/B]
fx = 2x - 4
fy = 2y

Critical point = (2,0)

The boundary of the triangle can be expressed in 3 lines, L1,L2, and L3. Find expressions for these lines.

L1 : x=0 y[-4,4]
L2 : y=-1/2x + 4 x[0,8]
L3 : y=1/2x - 4 x[0,8]

I checked the above, it is ALL correct.

Along L1, f can be expressed by the one variable function:
f = f(_,y) = ______
Along L2, f can be expressed by the one variable function:
f = f(x,_) = ______
Along L3, f can be expressed by the one variable function:
f = f(x,_) = ______

I don't know what to put in the blanks above
 
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  • #2
Chas3down said:

Homework Statement


Find the absolute maximum and minimum values of f(x,y) = y^2+x^2 -4x + 11 on the set D where D is the closed triangular region with vertices (8,0),(0,4) , and (0,-4) .

Homework Equations

The Attempt at a Solution


[/B]
fx = 2x - 4
fy = 2y

Critical point = (2,0)

The boundary of the triangle can be expressed in 3 lines, L1,L2, and L3. Find expressions for these lines.

L1 : x=0 y[-4,4]
L2 : y=-1/2x + 4 x[0,8]
L3 : y=1/2x - 4 x[0,8]

I checked the above, it is ALL correct.

Along L1, f can be expressed by the one variable function:
f = f(_,y) = ______
What's the x value along this vertical line?
Chas3down said:
Along L2, f can be expressed by the one variable function:
f = f(x,_) = ______
Given an x-value, how do you find the y value? You have the formula above.
Chas3down said:
Along L3, f can be expressed by the one variable function:
f = f(x,_) = ______
Given an x-value, how do you find the y value? You have the formula above
Chas3down said:
I don't know what to put in the blanks above
 
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Likes Chas3down
  • #3
Just plug in the formulas for x you have for the individual lines.
 
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Likes Chas3down
  • #4
Thanks, solved!
 

FAQ: Maximum and Minimum Values Inside Triangle

What is the maximum value inside a triangle?

The maximum value inside a triangle is the highest point or value that can be achieved within the triangle. It can refer to the maximum area, perimeter, or any other measurable quantity within the triangle.

How is the maximum value inside a triangle calculated?

The maximum value inside a triangle can be calculated using various mathematical formulas, depending on the specific quantity being measured. For example, the maximum area can be calculated using Heron's formula, while the maximum perimeter can be calculated using the triangle inequality theorem.

What factors affect the maximum value inside a triangle?

The maximum value inside a triangle is affected by the size and shape of the triangle. A larger triangle would generally have a higher maximum value compared to a smaller triangle. Additionally, the type of triangle (equilateral, isosceles, or scalene) and the placement of the maximum point within the triangle can also affect the maximum value.

What is the minimum value inside a triangle?

The minimum value inside a triangle is the lowest point or value that can be achieved within the triangle. Similar to the maximum value, it can refer to the minimum area, perimeter, or any other measurable quantity within the triangle.

How can the minimum value inside a triangle be determined?

The minimum value inside a triangle can be determined using various mathematical techniques, such as optimization methods or by finding the minimum value of a function that represents the quantity being measured. In some cases, the minimum value can also be found through geometric constructions or by considering the properties of the triangle.

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