The Lagrangian is
[tex]L(x,y,\lambda)=x^2-y^2+\lambda(x^2+y^2-81)[/tex]
with critical points where the partial derivatives are identically zero:
[tex]L_x=2x+2\lambda x=0\implies 2x(1+\lambda)=0\implies x=0\text{ or }\lambda=-1[/tex]
[tex]L_y=-2y+2\lambda y=0\implies-2y(1-\lambda)=0\implies y=0\text{ or }\lambda=1[/tex]
[tex]L_\lambda=x^2+y^2-81=0\implies x^2+y^2=81[/tex]
So we have four critical points to consider: (0, -9), (0, 9), (-9,0), and (9, 0). We have
[tex]f(0,-9)=-81[/tex]
[tex]f(0,9)=-81[/tex]
[tex]f(-9,0)=81[/tex]
[tex]f(9,0)=81[/tex]
So the maximum value is 81 and the minimum value is -81.
The extreme values of a function are the minimum and the maximum values of the function.
The extreme values are -81 and 81
The function is given as:
[tex]\mathbf{f(x,y) = x^2 - y^2}[/tex]
[tex]\mathbf{x^2 + y^2 = 81}[/tex]
Subtract 81 from both sides of [tex]\mathbf{x^2 + y^2 = 81}[/tex]
[tex]\mathbf{x^2 + y^2 - 81 = 0}[/tex]
Using Lagrange multiplies, we have:
[tex]\mathbf{L(x,y,\lambda) = f(x,y) + \lambda(0)}[/tex]
Substitute [tex]\mathbf{f(x,y) = x^2 - y^2}[/tex] and [tex]\mathbf{x^2 + y^2 - 81 = 0}[/tex]
[tex]\mathbf{L(x,y,\lambda) = x^2 - y^2 + \lambda(x^2 + y^2 - 81)}[/tex]
Differentiate
[tex]\mathbf{L_x = 2x + 2\lambda x}[/tex]
[tex]\mathbf{L_y = -2y + 2\lambda y}[/tex]
[tex]\mathbf{L_{\lambda} = x^2 + y^2 -81}[/tex]
Equate to 0
[tex]\mathbf{2x + 2\lambda x = 0}[/tex]
[tex]\mathbf{-2y + 2\lambda y = 0}[/tex]
[tex]\mathbf{x^2 + y^2 -81 = 0}[/tex]
So, we have:
[tex]\mathbf{2\lambda x = -2x}[/tex]
[tex]\mathbf{2\lambda y = 2y}[/tex]
Divide both sides of [tex]\mathbf{2\lambda x = -2x}[/tex] by -2x
[tex]\mathbf{\lambda = -1}[/tex]
Divide both sides of [tex]\mathbf{2\lambda y = 2y}[/tex] by 2y
[tex]\mathbf{\lambda = 1}[/tex]
The above means that:
[tex]\mathbf{\lambda = -1\ or\ x = 0}[/tex]
[tex]\mathbf{\lambda = 1\ or\ y = 0}[/tex]
Recall that: [tex]\mathbf{x^2 + y^2 = 81}[/tex]
When x = 0, we have:
[tex]\mathbf{0^2 + y^2 = 81}[/tex]
Take square roots of both sides
[tex]\mathbf{y = \±9}[/tex]
When y = 0, we have:
[tex]\mathbf{x^2 + 0^2 = 81}[/tex]
Take square roots of both sides
[tex]\mathbf{x = \±9}[/tex]
To determine the critical points, we consider:
[tex]\mathbf{\lambda = -1\ or\ x = 0}[/tex] or [tex]\mathbf{y = \±9}[/tex]
[tex]\mathbf{\lambda = 1\ or\ y = 0}[/tex] or [tex]\mathbf{x = \±9}[/tex]
So, the critical points are:
[tex]\mathbf{(x,y) = \{ (0, -9), (0, 9), (-9,0), (9, 0)\}}[/tex]
Substitute the above values in [tex]\mathbf{f(x,y) = x^2 - y^2}[/tex]
[tex]\mathbf{f(0,-9) = 0^2 - (-9)^2 = -81}[/tex]
[tex]\mathbf{f(0,-9) = 0^2 - (9)^2 = -81}[/tex]
[tex]\mathbf{f(-9,0) = (-9)^2 - 0^2 = 81}[/tex]
[tex]\mathbf{f(9,0) = (9)^2 - 0^2 = 81}[/tex]
Considering the above values, we have:
[tex]\mathbf{Minimum= -81}[/tex]
[tex]\mathbf{Maximum= 81}[/tex]
Hence, the extreme values are -81 and 81, respectively.
Read more about extreme values at;
brainly.com/question/1286349
