Consider the function f(x,y) = x^2 − 2xy + (y^3)/3 −3y

Question

(a) Compute all its critical points.

(b) Find Hessian matrices and quadratic forms for each critical point.

(c) For each quadratic form, compute the eigenvalues λ1, λ2 of the associated 2 × 2 matrix and find an orthogonal basis {w1,w2} of the corresponding eigenvectors.

(d) For each quadratic form sketch correct contour plots (with at least 3 labeled level sets) aligned with correct eigenlines.

(e) Based on eigenvalues and contour plots for each quadratic form classify each critical point of f as local minima, local maxima, or a saddle point.

Answer 1

This looks like a CALCULUS question

you must find the derivative of the function for x and y

set then equal to zero

then solve a system of equations

df/dx=2x-2y 2x-2y=0

df/dy=-2x+[3y²(3-3y)-y³(-3)]/(3-3y)2 all that =0 lets solve by substitution

solve for x on the first equation

2x-2y=0

2x=2y

x=y

substitute y for x on the 2nd equations -2y+[9y²-9y³+3y³ ]/(3-3y)²=0

multiply by (3-3y)² -2y(3-3y)² +9y²-6y³=0

square the binomial -2y(9-18y+9y²) +9y²-6y³ =0

multiply by -2y -18y+36y²-18y³+9y²-6y³=0

combine like terms and factor -24y³+45y²-18y=0

-3y(8y²-15y+6)=0

y=0 y= 15±√ 15²-4*8*6 / 2*8

y=(15+3√3)/16 and y=(15-3√3)/16

since x=y x=0 x=same x=same

so the critical points are (0,0) (15+..., 15+...) and (15-...,15-...)