Question

let ( f(x,y) ) be a function that has ( (-3,-7) ) as a critical point. we determine that ( f_{xx}(-3,-7)=8 ), ( f_{yy}(-3,-7)=1 ), and ( f_{xy}(-3,-7)=1 ). what does the d-test tell us about the function ( f )?
a. ( f ) has a relative maximum at ( (-3,-7) ).
b. ( f ) has a relative minimum at ( (-3,-7) ).
c. ( f ) has a saddle point at ( (-3,-7) ).
d. the answer cannot be determined from the information given.

Explanation:

Step1: Recall the second - derivative test formula

For a function \(z = f(x,y)\) with a critical point \((a,b)\), we calculate \(D=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}\). Here \(a=-3\), \(b = - 7\), \(f_{xx}(-3,-7)=8\), \(f_{yy}(-3,-7)=1\), and \(f_{xy}(-3,-7)=1\).

Step2: Calculate the discriminant \(D\)

Substitute the values into the formula: \(D=(8\times1)-(1)^{2}\).

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Since \(D=7>0\) and \(f_{xx}(-3,-7)=8>0\).

Answer:

B. f has a relative minimum at \((-3,-7)\)