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Consider a smooth function $f : \mathbb{R}^2 \to \mathbb{R}$, I wonder that any contour (curve) in $\mathbb{R}^2$ where every point of it is a local maxima of $f$, need be a smooth curve?

Edit : $f$ need to be smooth.

Edit 2 : By contour I mean curve of nonzero arc length.

Elaboration (after comments by Will and copper.hat)

Let the function be $f(x,y)$. I want the contour to have at every point on it, the $\frac{\partial f}{\partial x} = 0$ and $\frac{\partial^2 f}{\partial x^2} < 0$. Is any such contour which is not smooth possible?

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    @copper.hat : Thanks for the comment. The curve should be of non zero arc length.2012-10-24

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$ f(x,y) = - x^2 y^2 {}{}{}{}{} $

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    @RahulNarain : Thanks for the clarification. I guess its a rotation of axis by $45^o$ to form the new function.2012-10-24