The discriminant is zero at the origin for each of the

Chapter 13, Problem 44E

(choose chapter or problem)

The discriminant \(f_{x x} f_{y y}-f_{x y}^{2}\) is zero at the origin for each of the following functions, so the Second Derivative Test fails there. Determine whether the function has a maximum, a minimum, or neither at the origin by imagining what the surface \(z=f(x, y)\) looks like. Describe your reasoning in each case.

a. \(f(x, y)=x^{2} y^{2}\)

b. \(f(x, y)=1-x^{2} y^{2}\)

c. \(f(x, y)=x y^{2}\)

d. \(f(x, y)=x^{3} y^{2}\)

e. \(f(x, y)=x^{3} y^{3}\)

f. \(f(x, y)=x^{4} y^{4}\)

Equation Transcription:

   

    

Text Transcription:

f_xx f_yy-f^2xy

z=f(x,y)

f(x,y)=x^2y^2

f(x,y)=1-x^2y^2

f(x,y)=xy^2    

f(x,y)=x^3y^2

f(x,y)=x^3y^3  

f(x,y)=x^4y^4