let f(x,y) be a function that has (3, -6) as a critical point. we determine that f_xx(3, -6) = -5, f_yy(3…

let f(x,y) be a function that has (3, -6) as a critical point. we determine that f_xx(3, -6) = -5, f_yy(3, -6) = -9, and f_xy(3, -6) = 2. what does the d - test tell us about the function f?\na. f has a relative maximum at (3, -6).\nb. f has a relative minimum at (3, -6).\nc. f has a saddle point at (3, -6).\nd. the answer cannot be determined from the information given.

let f(x,y) be a function that has (3, -6) as a critical point. we determine that f_xx(3, -6) = -5, f_yy(3, -6) = -9, and f_xy(3, -6) = 2. what does the d - test tell us about the function f?\na. f has a relative maximum at (3, -6).\nb. f has a relative minimum at (3, -6).\nc. f has a saddle point at (3, -6).\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: Calculate the discriminant (D)

Given (f_{xx}(3, - 6)=-5), (f_{yy}(3, - 6)=-9), and (f_{xy}(3, - 6)=2). Substitute these values into the formula for (D): [ \begin{align*} D&=(-5)\times(-9)-(2)^{2}\ &=45 - 4\ &=41 \end{align*} ] Also, (f_{xx}(3, - 6)=-5<0)

Answer:

A. (f) has a relative maximum at ((3,-6))