Finding the maximum and minimum values

[Kork]

I am having trouble with getting started with this one:

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

I have started like this:
fx(x,y) = 1
fy(x,y) = 2

and then I am lost...How do I solve it?

 

[meldraft]

well, seeing as f(x,y)=0 is simply a line, you can find the intersection points between the line y=-1/2x, and the boundary of the disk, therefore the circle with radius=1.

If you consider f(x,y) as a scalar field, then it is a family of lines with incline=-1/2. Then, you just need to find the constant c (f(x,y)=c), for which the function is tangent to the boundary of the circle.

 

[HallsofIvy]

I am having trouble with getting started with this one:

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

I have started like this:
fx(x,y) = 1
fy(x,y) = 2

and then I am lost...How do I solve it?

Since those are never 0, there is no solution inside the circle. You need to look on the circle. There $y= \pm\sqrt{1- x^2}$ so f(x,y) becomes $x+ 2\sqrt{1-x^2}$ or $x- 2\sqrt{1- x^2}$. Differentiate those and see where the derivative is 0. Don't forget to look specifically at the value of f at the points (-1, 0) and (1, 0), the endpoints of those intervals.