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Find the local minimum for $f(x, y) = 2x^4 + y^2 - 4xy + 5y,\:x,y \in \mathbb{R}$ find the local minimum.

Okay this seems easy enough, the necessary condition dictates that candidates are of the form $\nabla f(x_0,y_0)=(0,0)$ but $\nabla f(x,y)=(8x^3-4y,2y-4x+5)$, so we end up with $4x^3-4x+5=0$. How this one solve this? Wolfram gives an estimate of -1.38 but is it possible to direstly find the root?

Thanks

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up vote 2 down vote accepted

The explicit expression for the unique real root of $4x^3 - 4x+5 = 0$ is not very elegant, so I do not think there is a nice way to obtain it, besides of just applying Cardano's formula

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Where did you get the equation $8x^3=4$ from? –  copper.hat Mar 22 '13 at 16:38
    
@copper.hat: from OP's typo –  Ilya Mar 22 '13 at 16:42
    
@Ilya my mistake –  bobdylan Mar 22 '13 at 17:41

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