Question

let ( f(x,y) ) be a function that has ( (-2,-4) ) as a critical point. we determine that ( f_{xx}(-2,-4)=-8 ), ( f_{yy}(-2,-4)=-9 ), and ( f_{xy}(-2,-4)=3 ). what does the d - test tell us about the function ( f )?
a. ( f ) has a relative maximum at ( (-2,-4) ).
b. ( f ) has a relative minimum at ( (-2,-4) ).
c. ( f ) has a saddle point at ( (-2,-4) ).
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 \(D=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}\). Given \(a=-2\), \(b = - 4\), \(f_{xx}(-2,-4)=-8\), \(f_{yy}(-2,-4)=-9\), and \(f_{xy}(-2,-4)=3\).

Step2: Calculate the discriminant \(D\)

$$
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$$

Step3: Analyze the second - derivative test results

Since \(D = 63>0\) and \(f_{xx}(-2,-4)=-8<0\). According to the second - derivative test: if \(D>0\) and \(f_{xx}(a,b)<0\) at a critical point \((a,b)\), then the function \(f(x,y)\) has a relative maximum at the point \((a,b)\).

Answer:

A. f has a relative maximum at \((-2,-4)\)