To determine whether a quadratic form has a maximum, minimum, or neither at the origin, we need to consider its associated matrix. Let Q be the matrix associated with the quadratic form. Then:
If Q is positive definite, the quadratic form has a minimum at the origin.
If Q is negative definite, the quadratic form has a maximum at the origin.
If Q has both positive and negative eigenvalues, the quadratic form has neither a maximum nor a minimum at the origin.
For the quadratic form 3x² - 4xy + y², the associated matrix is:
| 3 -2 |
|-2 1 |
The eigenvalues of this matrix are 2 and 1, which are both positive. Therefore, the quadratic form has a minimum at the origin.
For the quadratic form -4x² + xy + 5y², the associated matrix is:
|-4 1/2|
|1/2 5 |
The eigenvalues of this matrix are -3/2 and -5/2, which are both negative. Therefore, the quadratic form has a maximum at the origin.
For the quadratic form x² + 3xy + 4y², the associated matrix is:
|1 3/2|
|3/2 4 |
The eigenvalues of this matrix are 1/2 and 9/2, which are both positive. Therefore, the quadratic form has a minimum at the origin.
Therefore, the quadratic form 3x² - 4xy + y² has a minimum at the origin, the quadratic form -4x² + xy + 5y² has a maximum at the origin, and the quadratic form x² + 3xy + 4y² has a minimum at the origin.