- #1
mathmari
Gold Member
MHB
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Hey! ![Eek! :eek: :eek:](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Let and defined by with the euclidean norm .
Show that f has an uniquely defined global minimum and calculate it. I have done the following:
The partial derivative in respect to is
The gradient is
We get the critical point if we set the gradient equal to :
So, we have an extremum at .
The partial derivatives of second order are
So the Hessian-Matrix is:
Since we have a diagonal matrix, we get the eigenvalue from the diagonal: .
Since is positive, it follows that at the critical point we have a local minimum. Is everything correct? How can we show that this is a global minimum?
By uniquely defined is it mean that we have just one minimum?
(Wondering)
Let
Show that f has an uniquely defined global minimum and calculate it. I have done the following:
The gradient is
We get the critical point if we set the gradient equal to
So, we have an extremum at
The partial derivatives of second order are
So the Hessian-Matrix is:
Since we have a diagonal matrix, we get the eigenvalue from the diagonal:
Since
By uniquely defined is it mean that we have just one minimum?
(Wondering)