Question

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?
a. f has a relative maximum at (-8,-5).
b. f has a relative minimum at (-8,-5).
c. f has a saddle point at (-8,-5).
d. the answer cannot be determined from the information given.

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 \(D\):

$$
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$$

Since \(D=23>0\) and \(f_{xx}(-8,-5)=4>0\).

Answer:

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