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Second derivatives test :
If f have continuous partial derivatives on an open region containing a point for which
and .
To test for relative extrema of f , consider the quantity
1. If and , then f has a relative minimum at .
2. If and , then f has a relative maximum at .
3. If and then is a saddle point.
4. The test is inconclusive if .
The function is .
The domain is
Apply partial derivative on each side with respect to x.
Differentiate partially with respect to x.
Differentiate partially with respect to y.
The function is
Apply partial derivative on each side with respect to y
Differentiate partially with respect to y.
Differentiate partially with respect to x.
Find the critical points :
Equate to zero.
Equate to zero.
Substitute in equation (1).
Substitute in equation (1).
The critical points are and .
Find the value of f at the critical points :
Find the quantity D :
At the point .
Since and , the function f has a local minimum at .
Substitute the point in .
The local minimum is
At the point .
Since , the graph has saddle point at .
Find the value of f at the boundary points :
The domain of the function is .
Find the quantity D :
At the point .
Since , the graph has saddle point at .
At the point .
Since and , the function f has a local minimum at .
Substitute the point in .
The local minimum is
The local minimum is and
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