Multivariable calculus

Critical points

Where both partials vanish, and whether it is a max, a min or a saddle.

• Critical Points of x^2 + y^2 - 4x + 6yf(x,y) = x^2 + y^2 - 4x + 6y has one critical point, (2, -3), with discriminant D = 4 and f_xx = 2, so it is a local minimum of value -13.
• Critical Points of x^2 + xy + y^2 - 3yf(x,y) = x^2 + xy + y^2 - 3y has one critical point, (-1, 2). The cross term makes f_xy = 1, so D = 4 - 1 = 3 and the point is a local minimum.
• Critical Points of xy - x - yf(x,y) = xy - x - y has one critical point, (1, 1). Both pure second partials are zero, D = -1, and the point is a saddle.
• Critical Points of x^3 + y^3 - 3xyf(x,y) = x^3 + y^3 - 3xy has two critical points: (0,0) is a saddle with D = -9, and (1,1) is a local minimum with D = 27 and value -1.
• Critical Points of x^3 + y^3 - 3x - 3yf(x,y) = x^3 + y^3 - 3x - 3y has four critical points: a local min at (1,1), a local max at (-1,-1), and saddles at (1,-1) and (-1,1).
• Critical Points of x^4 + y^4 - 4xyf(x,y) = x^4 + y^4 - 4xy has three critical points: a saddle at (0,0) with D = -16 and local minima at (1,1) and (-1,-1) with D = 128.
• Critical Points of x^4 + y^4f(x,y) = x^4 + y^4 has one critical point, the origin, where D = 0. The second derivative test is inconclusive, but the origin is a global minimum.
• Critical Points of x^3 - 3xy^2f(x,y) = x^3 - 3xy^2 has one critical point, the origin, with D = 0. The test is inconclusive, but the surface is a monkey saddle there.
• Critical Points of x^2 + y^2 + x^2 yf(x,y) = x^2 + y^2 + x^2 y has three critical points: a local minimum at the origin with D = 4, and saddles at (sqrt 2, -1) and (-sqrt 2, -1).
• Critical Points of xy - x^2 y - xy^2f(x,y) = xy(1 - x - y) has four critical points: saddles at (0,0), (1,0) and (0,1), and a local maximum at (1/3, 1/3) with value 1/27.
• Critical Points of 2x^3 + xy^2 + 5x^2 + y^2f(x,y) = 2x^3 + xy^2 + 5x^2 + y^2 has four critical points: a local min at (0,0), a local max at (-5/3, 0), and saddles at (-1, 2) and (-1, -2).
• Critical Points of x^4 - 2x^2 + y^2f(x,y) = x^4 - 2x^2 + y^2 has three critical points: a saddle at the origin with D = -8, and global minima at (1,0) and (-1,0) with D = 16.
• Critical Points of sin(x) sin(y)Critical points of f(x,y) = sin(x) sin(y): a saddle at the origin with D = -1, a local max at (pi/2, pi/2) and a local min at (pi/2, -pi/2), both with D = 1.
• Critical Points of cos x + cos y + cos(x+y)Critical points of f(x,y) = cos x + cos y + cos(x+y): a local max at the origin with D = 3, a local min at (2pi/3, 2pi/3), and a saddle at (pi, pi).
• Critical Points of sin x + sin y + sin(x+y)Critical points of f(x,y) = sin x + sin y + sin(x+y): a local max at (pi/3, pi/3), a local min at (-pi/3, -pi/3), and a degenerate point at (pi, pi).
• Critical Points of sin(x+y) + cos(x-y)Critical points of f(x,y) = sin(x+y) + cos(x-y): a local max at (pi/4, pi/4) with D = 4, a saddle at (3pi/4, -pi/4), and a local min at (5pi/4, pi/4).
• Critical Points of cos(x) + y^2Critical points of f(x,y) = cos x + y^2 all lie on the x-axis: saddles at even multiples of pi with D = -2, local minima at odd multiples with D = 2.
• Critical Points of e^(xy)f(x,y) = e^(xy) has exactly one critical point, the origin, where D = -1. It is a saddle, because the exponential of a saddle surface is still a saddle.
• Critical Points of the Gaussian Bump e^(-x^2-y^2)The Gaussian bump f(x,y) = e^(-(x^2+y^2)) has exactly one critical point, the origin, where D = 4 and f_xx = -2: a local and global maximum of value 1.
• Critical Points of xy e^(-x^2-y^2)f(x,y) = xy e^(-x^2-y^2) has exactly five critical points: a saddle at the origin with D = -1, plus two maxima and two minima with D = 4/e^2.
• Critical Points of (x^2 + y^2) e^(-x)f(x,y) = (x^2+y^2)e^(-x) has exactly two critical points: a local minimum at the origin with D = 4, and a saddle at (2,0) with D = -4/e^4.
• Critical Points of sin(x) e^(-y^2)Critical points of f(x,y) = sin(x) e^(-y^2) all lie on the x-axis at odd multiples of pi/2, with D = 2: maxima where sin x = 1, minima where sin x = -1.