let f(x,y) be a function that has (9,7) as a critical point. we determine that f_xx(9,7)= - 5, f_yy(9,7)=…

let f(x,y) be a function that has (9,7) as a critical point. we determine that f_xx(9,7)= - 5, f_yy(9,7)= - 9, and f_xy(9,7)= - 6. what does the d - test tell us about the function f?\na. f has a relative maximum at (9,7).\nb. f has a relative minimum at (9,7).\nc. f has a saddle point at (9,7).\nd. the answer cannot be determined from the information given.

let f(x,y) be a function that has (9,7) as a critical point. we determine that f_xx(9,7)= - 5, f_yy(9,7)= - 9, and f_xy(9,7)= - 6. what does the d - test tell us about the function f?\na. f has a relative maximum at (9,7).\nb. f has a relative minimum at (9,7).\nc. f has a saddle point at (9,7).\nd. the answer cannot be determined from the information given.

Answer

Explanation:

Step1: Recall the formula for the second - derivative (D - test)

The formula for (D = f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}). Here, (a = 9), (b = 7), (f_{xx}(9,7)=-5), (f_{yy}(9,7)=-9), and (f_{xy}(9,7)=-6).

Step2: Calculate the value of (D)

Substitute the values into the formula: [ \begin{align*} D&=(-5)\times(-9)-(-6)^{2}\ &=45 - 36\ &=9 \end{align*} ] Also, (f_{xx}(9,7)=-5<0).

Step3: Use the second - derivative test criteria

If (D>0) and (f_{xx}(a,b)<0), then the function (f(x,y)) has a relative maximum at the point ((a,b)).

Answer:

A. f has a relative maximum at (9,7)