let f(x,y) be a function that has (-8,-5) as a critical point. we determine that f_xx(-8,-5)=4…

let f(x,y) be a function that has (-8,-5) as a critical point. we determine that f_xx(-8,-5)=4, f_yy(-8,-5)=8, and f_xy(-8,-5)=3. what does the d-test tell us about the function f?\na. f has a relative maximum at (-8,-5).\nb. f has a relative minimum at (-8,-5).\nc. f has a saddle point at (-8,-5).\nd. the answer cannot be determined from the information given.

let f(x,y) be a function that has (-8,-5) as a critical point. we determine that f_xx(-8,-5)=4, f_yy(-8,-5)=8, and f_xy(-8,-5)=3. what does the d-test tell us about the function f?\na. f has a relative maximum at (-8,-5).\nb. f has a relative minimum at (-8,-5).\nc. f has a saddle point at (-8,-5).\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}). Here, (a=-8), (b = - 5), (f_{xx}(-8,-5)=4), (f_{yy}(-8,-5)=8), and (f_{xy}(-8,-5)=3).

Step2: Calculate the discriminant (D)

Substitute the values into the formula (D): [ \begin{align*} D&=(4)\times(8)-(3)^{2}\ &=32 - 9\ &=23 \end{align*} ] Since (D=23>0) and (f_{xx}(-8,-5)=4>0).

Answer:

B. (f) has a relative minimum at ((-8,-5))