How Do You Determine the Extrema of f(x, y) = x + 2y on a Unit Disk?

[Kork]

Find the maximum and minimum values of f(x,y) = x+2y on the disk
x²+y^2 ≤1

I have this for now:

f_1(x,y) = 1
f_2(x,y) = 2

x=cos(t) and y=sin(t)

I have that g(t) = x(t) + 2*y(t) --> g(t) = cost(t) + 2*sin(t)

g'(t) = 0 = 2*cost-sin(t)

Then I can see that:

2cos(t)/cos(t) -sin(t)/cos(t) = 0/cos(t) --> tan = 2

That is the parameterization, right?

From this point I have no idea what to do.

 

[Ray Vickson]

Find the maximum and minimum values of f(x,y) = x+2y on the disk
x²+y^2 ≤1

I have this for now:

f_1(x,y) = 1
f_2(x,y) = 2

x=cos(t) and y=sin(t)

I have that g(t) = x(t) + 2*y(t) --> g(t) = cost(t) + 2*sin(t)

g'(t) = 0 = 2*cost-sin(t)

Then I can see that:

2cos(t)/cos(t) -sin(t)/cos(t) = 0/cos(t) --> tan = 2

That is the parameterization, right?

From this point I have no idea what to do.

Find the value or values of t that give tan(t) = 2. Or, since you only need sin(t) and cos(t), why not express them in terms of tan(t)?

RGV