How do I find critical points and determine local extrema for a given function?

[qq545282501]

Homework Statement

find all critical points and identify the locations of local maximums, minimums and saddle points of the function f(x,y)=4xy-\frac{x^4}{2}-y^2

Homework Equations

The Attempt at a Solution

setting Partial derivative respect to x = 0 : 4y-\frac{4x^3}{2}=0
setting partial derivative respect to y=0: 4x-2y=0

from the second equation, Y=2X, plug it into the first equation. I get : 2x(4-x^2)=0
now, x=0 or x=-2 or x=2. since y=2x, now I got 3 sets of critical points. (0,0), (-2,-4) and (2,4)

second partial derivative respect to x = -6x^2
second partial derivative respect to y=-2
and ƒ_xy=0

so D(a,b)= 12x^2

plug in these critical points into D(a,b) I found that local max value at (-2,-4) and (2,4) for which both give D value of 48, greater than 0, and ƒ_xx at both points are smaller than 0.
for D(0,0) , the test is inconclusive .

I did not find any local min, I am wondering if i missed something.
any help is appreciated

 

[ehild]

Homework Statement

find all critical points and identify the locations of local maximums, minimums and saddle points of the function f(x,y)=4xy-\frac{x^4}{2}-y^2

second partial derivative respect to x = -6x^2
second partial derivative respect to y=-2
and ƒ_xy=0

so D(a,b)= 12x^2

f_xy is not zero.

 

[Ray Vickson]

Homework Statement

find all critical points and identify the locations of local maximums, minimums and saddle points of the function f(x,y)=4xy-\frac{x^4}{2}-y^2

Homework Equations

The Attempt at a Solution

setting Partial derivative respect to x = 0 : 4y-\frac{4x^3}{2}=0
setting partial derivative respect to y=0: 4x-2y=0

from the second equation, Y=2X, plug it into the first equation. I get : 2x(4-x^2)=0
now, x=0 or x=-2 or x=2. since y=2x, now I got 3 sets of critical points. (0,0), (-2,-4) and (2,4)

second partial derivative respect to x = -6x^2
second partial derivative respect to y=-2
and ƒ_xy=0

so D(a,b)= 12x^2

plug in these critical points into D(a,b) I found that local max value at (-2,-4) and (2,4) for which both give D value of 48, greater than 0, and ƒ_xx at both points are smaller than 0.
for D(0,0) , the test is inconclusive .

I did not find any local min, I am wondering if i missed something.
any help is appreciated

$f_{xy} \neq 0$.

 

[qq545282501]

opps, got it. thank you