let ( f(x,y) ) be a function that has ( (-2,-4) ) as a critical point. we determine that ( f_{xx}(-2,-4)=-8…

let ( f(x,y) ) be a function that has ( (-2,-4) ) as a critical point. we determine that ( f_{xx}(-2,-4)=-8 ), ( f_{yy}(-2,-4)=-9 ), and ( f_{xy}(-2,-4)=3 ). what does the d - test tell us about the function ( f )?\na. ( f ) has a relative maximum at ( (-2,-4) ).\nb. ( f ) has a relative minimum at ( (-2,-4) ).\nc. ( f ) has a saddle point at ( (-2,-4) ).\nd. the answer cannot be determined from the information given.

let ( f(x,y) ) be a function that has ( (-2,-4) ) as a critical point. we determine that ( f_{xx}(-2,-4)=-8 ), ( f_{yy}(-2,-4)=-9 ), and ( f_{xy}(-2,-4)=3 ). what does the d - test tell us about the function ( f )?\na. ( f ) has a relative maximum at ( (-2,-4) ).\nb. ( f ) has a relative minimum at ( (-2,-4) ).\nc. ( f ) has a saddle point at ( (-2,-4) ).\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}). Given (a=-2), (b = - 4), (f_{xx}(-2,-4)=-8), (f_{yy}(-2,-4)=-9), and (f_{xy}(-2,-4)=3).

Step2: Calculate the discriminant (D)

[ \begin{align*} D&=f_{xx}(-2,-4)f_{yy}(-2,-4)-[f_{xy}(-2,-4)]^{2}\ &=(-8)\times(-9)-3^{2}\ &=72 - 9\ &=63 \end{align*} ]

Step3: Analyze the second - derivative test results

Since (D = 63>0) and (f_{xx}(-2,-4)=-8<0). According to the second - derivative test: if (D>0) and (f_{xx}(a,b)<0) at a critical point ((a,b)), then the function (f(x,y)) has a relative maximum at the point ((a,b)).

Answer:

A. f has a relative maximum at ((-2,-4))