let ( f(x,y) ) be a function that has ( (-3,-7) ) as a critical point. we determine that ( f_{xx}(-3,-7)=8…

let ( f(x,y) ) be a function that has ( (-3,-7) ) as a critical point. we determine that ( f_{xx}(-3,-7)=8 ), ( f_{yy}(-3,-7)=1 ), and ( f_{xy}(-3,-7)=1 ). what does the d-test tell us about the function ( f )?\na. ( f ) has a relative maximum at ( (-3,-7) ).\nb. ( f ) has a relative minimum at ( (-3,-7) ).\nc. ( f ) has a saddle point at ( (-3,-7) ).\nd. the answer cannot be determined from the information given.

let ( f(x,y) ) be a function that has ( (-3,-7) ) as a critical point. we determine that ( f_{xx}(-3,-7)=8 ), ( f_{yy}(-3,-7)=1 ), and ( f_{xy}(-3,-7)=1 ). what does the d-test tell us about the function ( f )?\na. ( f ) has a relative maximum at ( (-3,-7) ).\nb. ( f ) has a relative minimum at ( (-3,-7) ).\nc. ( f ) has a saddle point at ( (-3,-7) ).\nd. the answer cannot be determined from the information given.

Answer

Explanation:

Step1: Recall the second - derivative test formula

For a function (z = f(x,y)) with a critical point ((a,b)), we calculate (D=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}). Here (a=-3), (b = - 7), (f_{xx}(-3,-7)=8), (f_{yy}(-3,-7)=1), and (f_{xy}(-3,-7)=1).

Step2: Calculate the discriminant (D)

Substitute the values into the formula: (D=(8\times1)-(1)^{2}). [ \begin{align*} D&=8 - 1\ D&=7 \end{align*} ] Since (D=7>0) and (f_{xx}(-3,-7)=8>0).

Answer:

B. f has a relative minimum at ((-3,-7))