Is My Approach to Finding Max/Min on a Circular Disk Correct?

[Giuseppe]

Hey I was wondering if anyone can tell me if i am doing this right.

f(x,y) = xy^2 ; R is the circular disk x^2+y^2<=3

So first i took the gradient, since I know a mix/min can exist if the gradient is equal to 0.

Gradient of X= y^s
Gradient of Y= 2xy

So the point (0,0) can be considered right?

Anyway, I know I have to test region edges, so I parameterized the equation.

r(t) = <radical 3 cos(t),radical 3 sin(t)>

i plugged those values of x and y into my equation , and then took the derivative.

After some simplification, I came up with

sin(t)(3*radical3*cos(t)^2-3*radical3)

t= 0, pi, pi/2, 3pi/2 (right?)

so i found the x and y value when t is equal to those values

in conclusion, i have these points.

(0,0)
(0,radical3)
(0,-radical3)
(radical3,0)
(-radical3,0)

I tested these values in the equation xy^2 = f(x,y)

and found that there is no max or min...i don't think this is right.

Can someone help me find my mistake?

 

[member 553083]

Your mistake is in the derivative you took. The gradient of f(x,y) should be <y^2,2xy>. You should have gotten sin(t)(3*radical3*cos(t)^2+3*radical3)when you took the derivative. Then, when you plug in 0, pi, pi/2, 3pi/2 for t, you should get four critical points, two of which are maxima and two of which are minima.