Question

the function ( f ) has continuous second derivatives, and a critical point at ( (1,8) ). suppose ( f_{xx}(1,8)=-2, f_{xy}(1,8)=-2, f_{yy}(1,8)=2 ). then at the point ( (1,8) ):
a. there is a local minimum
b. there is a saddle point
c. the second derivative test is inconclusive
d. there is a local maximum
e. none of the above

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 \(f_{xx}(1,8)=- 2\), \(f_{xy}(1,8)=-2\), \(f_{yy}(1,8)=2\).
Substitute these values into the formula:
\(D=(-2)\times2-(-2)^{2}\)
\(D=-4 - 4\)
\(D=-8\)

Step2: Use the second - derivative test

The second - derivative test states that:

• If \(D>0\) and \(f_{xx}(a,b)>0\), then \(f(a,b)\) is a local minimum.
• If \(D>0\) and \(f_{xx}(a,b)<0\), then \(f(a,b)\) is a local maximum.
• If \(D<0\), then \((a,b)\) is a saddle point.
• If \(D = 0\), the test is inconclusive.

Since \(D=-8<0\)

Answer:

B. There is a saddle point