1. ## Extrema with constraints

Here is the question:

Find the extrema of $\displaystyle f(x, y) = x + y^2$, with constraint : $\displaystyle 2x^2 + y^2 = 1$

My attempts at solution:

I tried to solve it using Lagrange multiplier and I got:

$\displaystyle 1 = 4x \lambda$ ............................. I

$\displaystyle 2y = 2y \lambda$ ............................. II

$\displaystyle 2x^2 + y^2 = 1$ ............................. III

Now here is where I am stuck. How can I use the first two equations to get the value of either x or y?

2. Doesn't equation II tell you that $\displaystyle \lambda =1$ (assuming $\displaystyle y\ne 0 )$?

3. Originally Posted by Keep
Here is the question:

Find the extrema of $\displaystyle f(x, y) = x + y^2$, with constraint : $\displaystyle 2x^2 + y^2 = 1$

My attempts at solution:

I tried to solve it using Lagrange multiplier and I got:

$\displaystyle 1 = 4x lambda$ ............................. I

$\displaystyle 2y = 2y lambda$ ............................. II

$\displaystyle 2x^2 + y^2 = 1$ ............................. III

Now here is where I am stuck. How can I use the first two equations to get the value of either x or y?

$\displaystyle 2y-2y\lambda = 0$

$\displaystyle 2y(1 - \lambda) = 0$

$\displaystyle y = 0, \lambda = 1, x = \pm \frac{1}{\sqrt{2}}$

4. Your system of equations is correct.

From eq. II, $\displaystyle \lambda=1$
Plug that into eq. I, $\displaystyle x=\frac{1}{4}$
Plug $\displaystyle x=\frac{1}{4}$ into eq. III, $\displaystyle y=\pm\frac{1}{2}\sqrt\frac{7}{2}$

The other critical point occurs when $\displaystyle y=0$,
Use eq.III , $\displaystyle x=\frac{1}{\sqrt2}$

SO, your 4 critical points are...
$\displaystyle x=\frac{1}{4}$ , $\displaystyle y=\pm\frac{1}{2}\sqrt\frac{7}{2}$
$\displaystyle y=0$ , $\displaystyle x=\pm\frac{1}{\sqrt2}$

5. I'd use Calc 1.

Since $\displaystyle y^2=1-2x^2$ you only need to get the max's and min's of

$\displaystyle f(x)=x+1-2x^2$

6. Originally Posted by ojones
Doesn't equation II tell you that $\displaystyle \lambda =1$ (assuming $\displaystyle y\ne 0 )$?
Ooh OK I didn't look at it like that, I was trying to divide equation 1 by 2 and it wan't working. Yeah equation 2 says $\displaystyle \lambda$ is 1. Substitute that in equation 1 and we get x as 1/4. Substitute for x in equation III we get y = +/- $\displaystyle \sqrt {7/8}$

7. Originally Posted by Keep
Ooh OK I didn't look at it like that, I was trying to divide equation 1 by 2 and it wan't working. Yeah equation 2 says $\displaystyle \lambda$ is 1. Substitute that in equation 1 and we get x as 1/4. Substitute for x in equation III we get y = +/- $\displaystyle \sqrt {7/8}$
Sorry forgot to give you your other x point to try
$\displaystyle x = \frac{1}{4}$

8. So the maximum is $\displaystyle x=\frac{1}{4}$ , $\displaystyle y=\sqrt \frac{7}{8}$ and the minimum is $\displaystyle x=\frac{-1}{\sqrt2}, y=0$

### find the extrema with con

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