Say we have some arbitrary function $f(x,y) = xy$ or whatever. That doesn"t have to be a scalar-valued function (though for my concern it can be a restriction, no sure. Which is why I"m asking). If a problem asks to find the maximum and minimum values attained by that function along a particular path, what perform I should do? I recognize there room maximums and also minimums as soon as the derivative of a function is equal to $0$, however what does it average when you uncover maximums and minimums follow me a path? wgc2010.orgematically, exactly how do friend express a role along a path? A path in $\wgc2010.orgbb R^2$ is a continuous function $\phi : \to \wgc2010.orgbb R^2$. Recognize maxima and minima the a duty $f$ follow me a course $\phi$ way finding maxima and minima the the role $f \circ \phi$.

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For instance if $f(x, y) = xy$ and also $\phi(t) = (\cos(t), \sin(t))$, v $t \in <0, 2 \pi>$, climate to discover the maxima and also minima the $f$ along $\phi$ (the unit circle) girlfriend would find the maxima and minima the $$(f \circ \phi)(t) = \cos(t) \sin(t), \qquad t \in <0, 2\pi>$$

I recognize there room maximums and also minimums when the derivative the a role is equal to 0

This is no true, consider $f(x) = x^3$ in ~ $x = 0$. What is true is that if $f$ is differentiable in ~ $x_0$ and also $x_0$ is a regional extremum climate $f"(x_0) = 0$ Thanks because that contributing solution to wgc2010.orgematics stack Exchange!

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