Find the maximum and minimum values of the function f(x,y)=2x^2+3y^2–4x–5 on the domain x^2+y^2 less than or equal to 100. List points lexicographically.
I've discovered that the min = -7 [ f(1, 0) = -7).
And I know that the max = 299 but but I cannot get the correct points to match [ie: f( ?, ?) = 299].
Points for best answer, thanks for any help!!! :)
I've discovered that the min = -7 [ f(1, 0) = -7).
And I know that the max = 299 but but I cannot get the correct points to match [ie: f( ?, ?) = 299].
Points for best answer, thanks for any help!!! :)
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First, we find critical points inside the domain.
f_x = 4x - 4
f_y = 6y
Setting these equal to 0 yields (x, y) = (1, 0), which is inside x^2 + y^2 = 100.
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Now, we find points on the boundary of the circle (call it g) via Lagrange Multipliers.
grad(f) = λ grad(g)
==> <4x - 4, 6y> = λ <2x, 2y>.
Equate like entries:
4x - 4 = 2λx ==> 2x - 2 = λx
6y = 2λy ==> (λ - 3)y = 0 ==> λ = 3 or y = 0.
(i) If λ = 3, then 2x - 2 = 3x ==> x = -2.
Substituting this into x^2 + y^2 = 100 yields y = ±√96.
(ii) If y = 0, then substituting this into x^2 + y^2 = 100 yields x = ±10.
Testing all of the critical points:
f(1, 0) = -7 <----Minimum
f(-2, ±√96) = 299 <---Maximum
f(-10, 0) = 235
f(10, 0) = 155.
I hope this helps!
f_x = 4x - 4
f_y = 6y
Setting these equal to 0 yields (x, y) = (1, 0), which is inside x^2 + y^2 = 100.
-----------------------
Now, we find points on the boundary of the circle (call it g) via Lagrange Multipliers.
grad(f) = λ grad(g)
==> <4x - 4, 6y> = λ <2x, 2y>.
Equate like entries:
4x - 4 = 2λx ==> 2x - 2 = λx
6y = 2λy ==> (λ - 3)y = 0 ==> λ = 3 or y = 0.
(i) If λ = 3, then 2x - 2 = 3x ==> x = -2.
Substituting this into x^2 + y^2 = 100 yields y = ±√96.
(ii) If y = 0, then substituting this into x^2 + y^2 = 100 yields x = ±10.
Testing all of the critical points:
f(1, 0) = -7 <----Minimum
f(-2, ±√96) = 299 <---Maximum
f(-10, 0) = 235
f(10, 0) = 155.
I hope this helps!