To distinguish between minimum or maximum or saddle points one can use the second partial derivative test.
Let's define the Hessian matrix:
H(x,y)=(fxx(x,y)fyx(x,y)fxy(x,y)fyy(x,y)) ,
with partial derivatives of the function f(x,y) .
Let D(x,y) be the determinate of H(x,y) :
D(x,y)=fxx(x,y)fyy(x,y)−(fxy(x,y))2 .
If x=a,y=b - is critical point of function, then we have the following distinguish rule:
- If D(a,b)>0 and fxx(a,b)>0 then (a,b) is a local minimum of f(x,y) .
- If D(a,b)>0 and fxx(a,b)<0 then (a,b) is a local maximum of f(x,y) .
- If D(a,b)<0 then (a,b) is a saddle point of f(x,y) .
- If D(a,b)=0 then the second derivative test is inconclusive, and one should use Higher-order derivative test in order to classify the critical point.
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