Question

Find the absolute maximum of f(x,y) = 2x + 3y on the curve x^2 + 2y^2 = 1.

Answer 1

let x=sin(t) and y=cos(t)/sqrt(2). then x^2+2y^2=1.

Answer 2

now maximize 2sin(t)+3cos(t)/sqrt(2)

Answer 3

Where did you get those functions of x and y from?

Answer 4

start by letting x=sin(t) (trig substitution), then solve for y using the fact that x^2+2y^2=1
alternatively, you can let x=cos(t); either way, the answer will come out the same. we can set x equal to sin(t) since x must be between 0 and 1

Answer 5

sin(t)^2+2y(t)^2=1
2y(t)^2=1-sin(t)^2
sqrt(2)y(t)=sqrt(1-sin(t)^2)
sqrt(2)y(t)=cos(t)

Answer 6

you can also use lagrange multipliers to solve this problem, but trig substitution is a lot easier

Answer 7

*since x must be between -1 and 1

Answer 8

Oh I see. That makes sense, thank you!