LeGrange Multipliers Finding critical points of function

[Saladsamurai]

Homework Statement

Find the critical points of f(x,y)=x+y^2 subject to the constraint g(x)=x^2+y^2=1

Homework Equations

\nabla f=\lambda\nabla g
g(x,y)=1

The Attempt at a Solution

f_x=1=2\lambda*x\Rightarrow x=\frac{1}{2\lambda}

f_y=2y=2\lambda*y\Rightarrow y(\lambda-1)=0\Rightarrow y=0 \\ or\\ \lambda=1

x^2+y^2=1

I am a little confused as to where I go from here?

 

[HallsofIvy]

Homework Statement

Find the critical points of f(x,y)=x+y^2 subject to the constraint g(x)=x^2+y^2=1

Homework Equations

\nabla f=\lambda\nabla g
g(x,y)=1

The Attempt at a Solution

f_x=1=2\lambda*x\Rightarrow x=\frac{1}{2\lambda}

f_y=2y=2\lambda*y\Rightarrow y(\lambda-1)=0\Rightarrow y=0 \\ or\\ \lambda=1

x^2+y^2=1

I am a little confused as to where I go from here?

You've done the hard part. If y= 0, then \lambda doesn't matter: Find x from x²+ y²= 1. If \lambda= 1, then x= 1/2 and you can find y from x²+ y²= 1.