Absolute Max and Min

GROUP MEMBERS:

1. _____________________________

2. ______________________________

3. ______________________________

4. ______________________________

Goal: To find the absolute maximum and absolute minimum of the function

f(x,y) = x2+ y2-xy+y on the triangular region D whose vertices are the points (0,0), (-1,0) and (0, -1). Recall that the function has a single critical point located at (-1/3, -2/3) and that this point is a local minimum.

Person 1: Make a sketch of the region D.

 

 

 

 

 

 

 

 

This boundary (edge) of this region consists of three line segments. Each person will study one of these line segments and then work out the following on your own sheet (see following pages).

Person 1: the line segment l1 from (0,0) to (-1,0)

Person 2 and Person 3: the line segment l2 from (-1,0) to (0, -1)

Person 4: the line segment l3 from (0,-1) to (0, 0).

Before starting your individual work, please read through these directions together.

a. Give parametric equations for the lines: x(t), y(t), for t in (tmin, tmax).

Recall l(t) = pt + t vector. What are the starting and finishing values of t for your parameterization? What are the (x,y) coordinates of these two points? When you have done this, write your answers here, show your method to your group and explain.

l1(t) = ( x(t) = -t , y(t) = 0 ), t in [0 , 1 ]. Endpoints = (0,0) and (-1,0)

l2(t) = ( x(t) = -1 +t , y(t) = -t ), t in [0, 1]. Endpoints = (-1,0), (0,1)

l2(t) = ( x(t) =0 , y(t) = -1 + t ), t in [0, 1]. Endpoints = (0, -1) and (0,0).

  

b. Having determined (x(t), y(t)), you can now determine the value of the function f(x,y) = f(x(t), y(t)) = f(t) along the edge. The function is now a function of a single variable t.

Determine the maximum and minimum of f(t) for t in [tmin, tmax].

    1. First find the critical point of the function f(t) (if it has any).
    2. Find the value of the function at the critical point(s).
    3. Find the value of the function at the endpoints of the t interval:

      4. f(tmin) and f(tmax).

    4. Comparing the values at the critical point and the endpoints, find the maximum and minimum along your edge.

 

When you are done with your calculations, record your values here and explain your results to the group:

a. For l1: f(t)= t2 for t in [ 0,1 ].

Critical point at t = 0 . Value of function at critical point = 0 .

Values of function at endpoints of t interval = f(0) = 0 , f(1) = 1 .

Maximum of f on l1= 1 and occurs at point ( x =-1, y = 0).

Minimum of f on l1 = 0 and occurs at point ( x =0, y = 0).

b. For l2: f(t)= 3t2 - 4t + 1 for t in [0, 1].

Critical point at t = 2/3 . Value of function at critical point =-1/3 .

Values of function at endpoints of t interval = 1 at t=0, =0 at t=1.

Maximum of f on l2= 1 and occurs at point ( x =-1, y =0 )

Minimum of f on l2 = -1/3 and occurs at point ( x = -1/3, y =-2/3)

c. For l3: f(t)= t2 - t for t in [0, 1].

Critical point at t = 1/2 . Value of function at critical point = -1/4 .

Values of function at endpoints of t interval =0 at both endpoints.

Maximum of f on l3= 0 and occurs at point ( x =-1, y = 0 ) and at (0,0)

Minimum of f on l3 = 3/4 and occurs at point ( x = 0 , y = -1/2)

 

d. Looking at all 3 edges, the maximum of f on boundary = 1 occurs at point = (-1,0)

the minimum of f on the boundary = -1/3 occurs at point = (-1/3, -2/3)

e. The value of f at the critical point is f(-1/3, -2/3) = -1/3

f. Comparing the values of the function at the critical point with the values on the boundary,

the absolute maximum of f over the region D = 1 occurs at (x,y) = (-1,0)

the absolute minimum of f over the region D = -1/3 occurs at (x,y) = (-1/3, - 2/3)

 

Person 1 studying line 1:

a. Give parametric equations for your line segment: x(t), y(t), for t in (tmin, tmax).

Recall l(t) = pt + t vector.

 

 l(t) = (0,0) + t( -1, 0) = (-t, 0)

x(t) = -t

y(t) = 0

 

What are the starting and finishing values of t for your parameterization?

 t \in [0,1]

 

What are the (x,y) coordinates of these two points?

l(0) = (0,0)

l(1)= (-1,0)

 

When you have done this, write your answers here, show your method to your group and explain.

 

l1(t) = ( x(t) = -t , y(t) = 0 ), t in [0 , 1 ]. Endpoints = (0,0) and (-1,0)

 

 

b. Having determined (x(t), y(t)), you can now determine the value of the function f(x,y) = f(x(t), y(t)) = f(t) along the edge. The function is now a function of a single variable t.

f(x,y) = x2+ y2-xy+y

Determine the maximum and minimum of f(t) for t in [tmin, tmax].

    1. f(t) = t2 , t in [0,1]

      2.  

    2. First find the critical point of the function f(t) (if it has any).

      3. f'(t) = 2t = 0 at t = 0

       

    3. Find the value of the function at the critical point(s).

      4.  

       f(0) = 0

       

    4. Find the value of the function at the endpoints of the t interval: f(tmin) and f(tmax).

      5. f(0) = 0

      f(1) = 1

       

    5. Comparing the values at the critical point and the endpoints, find the maximum and minimum along your edge.

      6.  

       max at t=1; f(1) =0

      min at t = 0; f(0) = 0

       

      Record your results on your team's master list.

       

      Person 2 studying line 2:

      a. Give parametric equations for your line segment: x(t), y(t), for t in (tmin, tmax).

      Recall l(t) = pt + t vector.

       l(t) = (-1,0) + t ( 1, -1) = ( -1 + t, -t)

       

      x(t) = -1 + t

      y(t) = -t

       

      What are the starting and finishing values of t for your parameterization?

       

       t in [0, 1]

       

       

      What are the (x,y) coordinates of these two points?

       l(0) = (-1,0)

      l(1) = (0, -1)

      When you have done this, write your answers here, show your method to your group and explain.

       

      l2(t) = ( x(t) = -1 +t , y(t) = -t ), t in [0, 1]. Endpoints = (-1,0), (0,1)

       

       

      b. Having determined (x(t), y(t)), you can now determine the value of the function f(x,y) = f(x(t), y(t)) = f(t) along the edge. The function is now a function of a single variable t.

       

      f(x,y) = x2+ y2-xy+y

      Determine the maximum and minimum of f(t) for t in [tmin, tmax].

    6. f(t) = (-1 + t)2 + t2 -(-1 + t)(-t) + (-t) = 3t2 - 4t + 1

      7.  

    7. First find the critical point of the function f(t) (if it has any).

      8. f'(t) = 6t - 4 = 0 then t = 2/3

       

       

    8. Find the value of the function at the critical point(s).

      9.  

       f(2/3) = -1/3

       

    9. Find the value of the function at the endpoints of the t interval: f(tmin) and f(tmax).

      10. f(0) = 1

      f(1) = 0

       

       

    10. Comparing the values at the critical point and the endpoints, find the maximum and minimum along your edge.

      11.  

       min = -1/3 occurs at t = 2/3 corresponds to (x = -1/3, y = -2/3).

       

       

      Record your results on your team's master list.

       

      Person 4 studying line 3:

      a. Give parametric equations for your line segment: x(t), y(t), for t in (tmin, tmax).

      Recall l(t) = pt + t vector.

       l(t0 = (0,-1) + t (0, 1) = (0, -1 + t)

      x(t) = 0

      y(t) = -1 + t

       

      What are the starting and finishing values of t for your parameterization?

       

       t in [0,1]

       

      What are the (x,y) coordinates of these two points?

        at t =0, (0, -1)

      at t = 1, (0,0)

      When you have done this, write your answers here, show your method to your group and explain.

       

      l3(t) = ( x(t) =0 , y(t) = -1 + t ), t in [0, 1]. Endpoints = (0, -1) and (0,0).

       

       

      b. Having determined (x(t), y(t)), you can now determine the value of the function f(x,y) = f(x(t), y(t)) = f(t) along the edge. The function is now a function of a single variable t.

      f(x,y) = x2+ y2-xy+y

      Determine the maximum and minimum of f(t) for t in [tmin, tmax].

    11. f(t) = (-1 +t)2 + (-1 + t)= t2 - t

      12.  

    12. First find the critical point of the function f(t) (if it has any).

      13. f'(t) = 2t -1 = 0 at t = 1/2

       

       

    13. Find the value of the function at the critical point(s).

      14.  

       f(1/2) = 1/4 - 1/2 = - 1/4

       

    14. Find the value of the function at the endpoints of the t interval: f(tmin) and f(tmax).

      15. f(0) = 0

      f(1) = 0

    15. Comparing the values at the critical point and the endpoints, find the maximum and minimum along your edge.

max = 0 at t =0 and t=1 correspondng to (-1,0) and (0,0)

min = -1/4 at t = 1/2 corresponding to (0, -1/2).

 

 

 

Record your results on your team's master list.