let ( f(x,y) ) be a function that has ( (-8,0) ) as a critical point. we determine that ( f_{xx}(-8,0)=9…

let ( f(x,y) ) be a function that has ( (-8,0) ) as a critical point. we determine that ( f_{xx}(-8,0)=9, f_{yy}(-8,0)=8 ), and ( f_{xy}(-8,0)=-2 ). what does the d - test tell us about the function ( f )?\na. ( f ) has a relative maximum at ( (-8,0) ).\nb. ( f ) has a relative minimum at ( (-8,0) ).\nc. ( f ) has a saddle point at ( (-8,0) ).\nd. the answer cannot be determined from the information given.

let ( f(x,y) ) be a function that has ( (-8,0) ) as a critical point. we determine that ( f_{xx}(-8,0)=9, f_{yy}(-8,0)=8 ), and ( f_{xy}(-8,0)=-2 ). what does the d - test tell us about the function ( f )?\na. ( f ) has a relative maximum at ( (-8,0) ).\nb. ( f ) has a relative minimum at ( (-8,0) ).\nc. ( f ) has a saddle point at ( (-8,0) ).\nd. the answer cannot be determined from the information given.

Answer

Explanation:

Step1: Calculate the discriminant (D)

The formula for the discriminant (D) of a function (f(x,y)) at a critical point ((a,b)) is (D = f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}). Given (f_{xx}(- 8,0)=9), (f_{yy}(-8,0)=8), and (f_{xy}(-8,0)=-2). Substitute these values into the formula: (D=(9\times8)-(-2)^{2}).

Step2: Simplify the expression for (D)

First, calculate (9\times8 = 72) and ((-2)^{2}=4). Then (D = 72 - 4=68). Since (D>0) and (f_{xx}(-8,0)=9>0).

Answer:

B. (f) has a relative minimum at ((-8,0))