Question

let ( f(x,y) ) be a function that has ( (-8,0) ) as a critical point. we determine that ( f_{xx}(-8,0)=9, f_{yy}(-8,0)=8 ), and ( f_{xy}(-8,0)=-2 ). what does the d - test tell us about the function ( f )?
a. ( f ) has a relative maximum at ( (-8,0) ).
b. ( f ) has a relative minimum at ( (-8,0) ).
c. ( f ) has a saddle point at ( (-8,0) ).
d. the answer cannot be determined from the information given.

Explanation:

Step1: Calculate the discriminant \(D\)

The formula for the discriminant \(D\) of a function \(f(x,y)\) at a critical point \((a,b)\) is \(D = f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}\).
Given \(f_{xx}(- 8,0)=9\), \(f_{yy}(-8,0)=8\), and \(f_{xy}(-8,0)=-2\).
Substitute these values into the formula: \(D=(9\times8)-(-2)^{2}\).

Step2: Simplify the expression for \(D\)

First, calculate \(9\times8 = 72\) and \((-2)^{2}=4\).
Then \(D = 72 - 4=68\).
Since \(D>0\) and \(f_{xx}(-8,0)=9>0\).

Answer:

B. \(f\) has a relative minimum at \((-8,0)\)