Question

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?
a. f has a relative maximum at (9,7).
b. f has a relative minimum at (9,7).
c. f has a saddle point at (9,7).
d. the answer cannot be determined from the information given.

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:

$$
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$$

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)