find the local maxima, local minima and saddle point of the function f(x,y)=2x3 +2y3 -9x2 +3y2-12y

Question

find the local maxima, local minima and saddle point of the function f(x,y)=2x3 +2y3 -9x2 +3y2-12y

Answer 1

f(x,y) = 2x³ + 2y³ - 9x² + 3y² - 12y

 

The critical points are points where ∇f(x,y) = 0:

 

6x² - 18x = 0 ⇒ 6x(x-3) = 0 ⇒ x = 0 or 3

 

6y² + 6y - 12 = 0 ⇒ 6(y + 2)(y - 1) = 0 ⇒ y = -2 or 1

 

So there are four critical points: (0,-2), (0,1), (3,-2), (3,1)

 

Let's use the Hessian Criterion.

 

H f(x,y) =

 

⌈ 12x-18     0     ⌉

⌊     0      12y+6 ⌋

 

 

 

H f(0,-2) =

 

⌈ -18   0 ⌉
⌊ 0   -18 ⌋ (Negative Definite)

 

 

H f(0,1)

⌈ -18   0 ⌉
⌊   0  18 ⌋ (Indefinite)

 

 

H f(3,-2) =

⌈ 18   0 ⌉
⌊  0 -18 ⌋ (Indefinite)

 

 

H f(3,1) =

⌈ 18   0 ⌉
⌊  0  18 ⌋ (Positive Definite)

 

This shows that (0,1) and (3,-2) are saddle points. f(0,1) = f(3,-2) = -7

 

(3,1) is local minimum point. f(3,1) = -34

 

(0,-2) is the site of a local maximum point. f(0,-2) = 20