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.