Critical Points of f(x,y)= x^{3} - 6xy + y^{3}

jegues
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Homework Statement



Find and classify all critical points of,

f(x,y)= x^{3} - 6xy + y^{3}

Homework Equations





The Attempt at a Solution



See figure attached for my attempt at the solving all the critical points. I'm getting some weird numbers so I want to make sure I did this part correct before I start classifying them.

Does anyone see any problems?

Thanks again!
 

Attachments

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hi jegues! :smile:

in line 6, your 1/8 should be 1/4 :redface:

(btw, it would be a lot easier if you divided everything by 3 first)
 
You can quickly check your work using wolfram alpha... and according to it, (0,0) is one but your other answers are wrong.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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