Conditional extreme values
Can there be two maximum points of a #-Link-Snipped-# with 2 variables?
For example, this question:
Find the extreme values (state whether it's maximum or minimum) of f(x,y)= x^2 + 2xy + 7y^2 under the terms of x^2 + 4y^2 = 5
I got 2 minimum values at f(1,1)= 10 and at f(-1,-1)=10. Turns out that the values at both points are the same. Can this be correct?
Here's my working:
Let f(x,y)= x^2 + 2xy + 7y^2 with constraint g(x,y) = x^2 + 4y^2 = 5.
Then, ∇f = λ∇g ==> <2x + 2y, 2y + 14y> = λ<2x, 8y>.
Equate like entries:
2x + 2y = 2λx ==> x + y = λx
2y + 14y = 2λy ==> 2 = λ
So, x + y = λx
==> y = x.
Substitute this into g ==> x^2+4x^2=5 and thus, x= 1 or -1
Hence, (x,y) = (1,1) and (-1,-1)
D(1,1) = (-2)*(32) + (-14)*(16)= -288 <0
D(-1,-1)=(-2)*(32)+(-4)(0)=-64<0
SO,the maximum value at f(1,1)=10 and at f(-1,-1)=10
For example, this question:
Find the extreme values (state whether it's maximum or minimum) of f(x,y)= x^2 + 2xy + 7y^2 under the terms of x^2 + 4y^2 = 5
I got 2 minimum values at f(1,1)= 10 and at f(-1,-1)=10. Turns out that the values at both points are the same. Can this be correct?
Here's my working:
Let f(x,y)= x^2 + 2xy + 7y^2 with constraint g(x,y) = x^2 + 4y^2 = 5.
Then, ∇f = λ∇g ==> <2x + 2y, 2y + 14y> = λ<2x, 8y>.
Equate like entries:
2x + 2y = 2λx ==> x + y = λx
2y + 14y = 2λy ==> 2 = λ
So, x + y = λx
==> y = x.
Substitute this into g ==> x^2+4x^2=5 and thus, x= 1 or -1
Hence, (x,y) = (1,1) and (-1,-1)
D(1,1) = (-2)*(32) + (-14)*(16)= -288 <0
D(-1,-1)=(-2)*(32)+(-4)(0)=-64<0
SO,the maximum value at f(1,1)=10 and at f(-1,-1)=10
0
