let ( f(x,y) ) be a function that has ( (1,-1) ) as a critical point. we determine that ( f_{xx}(1,-1)=-2 )…

let ( f(x,y) ) be a function that has ( (1,-1) ) as a critical point. we determine that ( f_{xx}(1,-1)=-2 ), ( f_{yy}(1,-1)=-4 ), and ( f_{xy}(1,-1)=-1 ). what does the d - test tell us about the function ( f )?\na. ( f ) has a relative maximum at ( (1,-1) ).\nb. ( f ) has a relative minimum at ( (1,-1) ).\nc. ( f ) has a saddle point at ( (1,-1) ).\nd. the answer cannot be determined from the information given.

let ( f(x,y) ) be a function that has ( (1,-1) ) as a critical point. we determine that ( f_{xx}(1,-1)=-2 ), ( f_{yy}(1,-1)=-4 ), and ( f_{xy}(1,-1)=-1 ). what does the d - test tell us about the function ( f )?\na. ( f ) has a relative maximum at ( (1,-1) ).\nb. ( f ) has a relative minimum at ( (1,-1) ).\nc. ( f ) has a saddle point at ( (1,-1) ).\nd. the answer cannot be determined from the information given.

Answer

Explanation:

Step1: Recall the second - derivative test formula

The second - derivative test for a function (z = f(x,y)) at a critical point ((a,b)) uses the discriminant (D=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}).

Step2: Substitute the given values

Given (f_{xx}(1, - 1)=-2), (f_{yy}(1, - 1)=-4), and (f_{xy}(1, - 1)=-1). Then (D=(-2)\times(-4)-(-1)^{2}).

First, calculate ((-2)\times(-4)=8) and ((-1)^{2}=1).

So (D = 8 - 1=7).

Also, (f_{xx}(1,-1)=-2<0).

Step3: Apply the second - derivative test rules

If (D>0) and (f_{xx}(a,b)<0), then the function (f(x,y)) has a relative maximum at the point ((a,b)).

Since (D = 7>0) and (f_{xx}(1,-1)=-2<0), the function (f) has a relative maximum at ((1,-1))

Answer:

A. f has a relative maximum at ((1,-1))